Integer Exponents and Scientific Notation

Product and Quotient Rules for Exponents

When you see an expression like \(2^4\), the base is \(2\) and the exponent is \(4\). The exponent tells how many times the base is used as a factor:

$$2^4 = 2 \cdot 2 \cdot 2 \cdot 2$$

Exponents are a shortcut for repeated multiplication. In this lesson, you will learn what happens when you multiply or divide powers that have the same base.

These rules are very useful when working with large numbers, small numbers, and scientific notation.

1. Product Rule for Exponents

When multiplying powers with the same base, you add the exponents.

$$a^m \cdot a^n = a^{m+n}$$

This rule only works when the bases are the same.

Why does this work? Expand the powers:

$$a^m \cdot a^n = \underbrace{a \cdot a \cdot a \cdots a}_{m\text{ factors}} \cdot \underbrace{a \cdot a \cdot a \cdots a}_{n\text{ factors}}$$

Now all the factors are together. That gives a total of \(m+n\) factors of \(a\), so the result is \(a^{m+n}\).

• \(x^3 \cdot x^2 = x^{3+2} = x^5\)
• \(5^4 \cdot 5^1 = 5^{4+1} = 5^5\)
• \(10^2 \cdot 10^6 = 10^{8}\)

Important: Do not add exponents if the bases are different.

• \(2^3 \cdot 3^3\) cannot become \(6^6\)
• \(x^2 \cdot y^2\) cannot become \((xy)^4\)

If the bases are different, the product rule does not apply.

2. Quotient Rule for Exponents

When dividing powers with the same base, you subtract the exponents.

$$\frac{a^m}{a^n} = a^{m-n}$$

This also only works when the bases are the same, and the base cannot be zero.

Why does this work? Write out the factors:

$$\frac{a^5}{a^2} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a}$$

Two \(a\)'s in the numerator and denominator cancel, leaving:

$$a \cdot a \cdot a = a^3$$

Since \(5-2=3\), this matches the rule:

$$\frac{a^5}{a^2} = a^{5-2} = a^3$$

• \(\frac{x^7}{x^4} = x^{7-4} = x^3\)
• \(\frac{10^9}{10^3} = 10^{9-3} = 10^6\)
• \(\frac{m^5}{m^5} = m^{5-5} = m^0 = 1\)

3. Connecting the Rules to Repeated Multiplication

The product rule and quotient rule come from counting factors.

For multiplication, you are combining groups of the same factor.

For division, you are removing matching factors from the top and bottom.

This is why:

• Multiply same base → add exponents
• Divide same base → subtract exponents

4. Worked Examples

Example 1: Multiply powers with the same base

Simplify: \(y^4 \cdot y^3\)

Step 1: Check the bases. Both bases are \(y\), so use the product rule.

Step 2: Add the exponents.

$$y^4 \cdot y^3 = y^{4+3} = y^7$$

Answer: \(y^7\)

Example 2: Divide powers with the same base

Simplify: \(\frac{n^8}{n^2}\)

Step 1: The bases are both \(n\), so use the quotient rule.

Step 2: Subtract the exponents.

$$\frac{n^8}{n^2} = n^{8-2} = n^6$$

Answer: \(n^6\)

Example 3: Use coefficients and exponents together

Simplify: \(3x^2 \cdot 4x^5\)

Step 1: Multiply the numbers in front: \(3 \cdot 4 = 12\).

Step 2: Multiply the powers with the same base \(x\), so add exponents.

$$3x^2 \cdot 4x^5 = 12x^{2+5} = 12x^7$$

Answer: \(12x^7\)

Example 4: Divide a coefficient and a power

Simplify: \(\frac{18a^6}{3a^2}\)

Step 1: Divide the coefficients: \(18 \div 3 = 6\).

Step 2: Divide the powers with the same base \(a\), so subtract exponents.

$$\frac{18a^6}{3a^2} = 6a^{6-2} = 6a^4$$

Answer: \(6a^4\)

5. Common Mistakes to Avoid

• Do not add exponents when dividing.
  \(\frac{x^5}{x^2} = x^3\), not \(x^7\).
• Do not subtract exponents when multiplying.
  \(x^5 \cdot x^2 = x^7\), not \(x^3\).
• Make sure the bases are the same.
  \(x^3 \cdot y^3\) does not become \((xy)^6\).
• Do not forget the coefficient.
  In \(2m^3 \cdot 5m^2\), the numbers multiply too: \(2 \cdot 5 = 10\).

6. Quick Practice Ideas

Try these on your own:

1. \(b^2 \cdot b^6\)
2. \(\frac{p^9}{p^4}\)
3. \(7r^3 \cdot 2r^4\)
4. \(\frac{20t^5}{5t^2}\)

Answers:

1. \(b^8\)
2. \(p^5\)
3. \(14r^7\)
4. \(4t^3\)

7. Summary

The product rule says that when you multiply powers with the same base, you add the exponents:

$$a^m \cdot a^n = a^{m+n}$$

The quotient rule says that when you divide powers with the same base, you subtract the exponents:

$$\frac{a^m}{a^n} = a^{m-n}$$

Always check that the bases match before using these rules. If you remember multiply = add exponents and divide = subtract exponents, you will be able to simplify many exponent expressions correctly.

Power of a Power and Power of a Product

When working with exponents, there are patterns that help us simplify expressions quickly and correctly. In this lesson, you will learn two important exponent rules: power of a power and power of a product.

These rules are useful because they help you rewrite expressions without multiplying everything out every time. They also connect to the meaning of exponents as repeated multiplication.

First, remember what an exponent means. In an expression like \(3^4\), the base is 3 and the exponent is 4. This means:

The exponent tells how many times the base is used as a factor.

Rule 1: Power of a Power

A power of a power happens when one exponent is raised to another exponent, like \((2^3)^4\).

To simplify a power of a power, multiply the exponents:

This rule works because the first exponent tells how many copies of the base are in each group, and the second exponent tells how many groups there are.

For example:

First, write \(2^3\) as repeated multiplication:

This means there are 4 groups of \(2 \cdot 2 \cdot 2\):

Now count the total number of 2s. There are \(3 \times 4 = 12\) factors of 2.

So instead of multiplying the base exponents separately, we multiply the exponents together.

Rule 2: Power of a Product

A power of a product happens when a product is raised to an exponent, like \((3 \cdot 5)^2\) or \((ab)^4\).

To simplify a power of a product, apply the exponent to each factor:

This rule works because the whole product is being multiplied by itself several times.

For example:

Now reorder the factors:

So:

This means you can raise each factor in the product to the same exponent.

Important idea: These rules come from expanding expressions. If you are unsure, write the expression out as repeated multiplication and look for the pattern.

Worked Example 1

Simplify \((5^2)^3\).

Use the power of a power rule. Multiply the exponents:

Answer: \(5^6\)

Worked Example 2

Simplify \((2x)^4\).

Use the power of a product rule. Apply the exponent to both factors:

Now simplify \(2^4\):

So the expression becomes:

Answer: \(16x^4\)

Worked Example 3

Simplify \(((3^2)^2)\).

This is a power of a power. Multiply the exponents:

If you want to evaluate it fully:

Answer: \(3^4\), or \(81\)

Worked Example 4

Simplify \((2a^3b)^2\).

This expression is a product of three factors: \(2\), \(a^3\), and \(b\). Apply the exponent to each factor:

Now use the power of a power rule on \((a^3)^2\):

Also simplify \(2^2\):

So the final answer is:

Answer: \(4a^6b^2\)

Common Mistakes to Avoid

• Do not add exponents in a power of a power. For example, \((x^2)^3\) is \(x^6\), not \(x^5\).

• Do not forget to apply the exponent to every factor in a product. For example, \((2x)^3 = 2^3x^3 = 8x^3\), not \(2x^3\).

• Keep track of parentheses. Parentheses show what the exponent applies to. In \((ab)^2\), the 2 applies to both \(a\) and \(b\).

Quick Check

1. Simplify \((y^4)^2\)

2. Simplify \((3m)^2\)

3. Simplify \((x^2z)^3\)

Answers

1. \((y^4)^2 = y^8\)

2. \((3m)^2 = 3^2m^2 = 9m^2\)

3. \((x^2z)^3 = (x^2)^3z^3 = x^6z^3\)

Summary

There are two main rules in this lesson:

• Power of a power: multiply the exponents.

• Power of a product: apply the exponent to each factor.

In symbols:

If you forget a rule, go back to the meaning of exponents as repeated multiplication. Expanding the expression can help you see why the rule works.

Zero and Negative Exponents

Exponents tell us how many times a number is multiplied by itself. For example, \(3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\).

In this lesson, you will learn what happens when the exponent is zero or negative. These kinds of exponents may look strange at first, but they follow the same exponent patterns you already know.

By the end, you should understand two important ideas:

• Any non-zero number raised to the zero power equals 1.
• A negative exponent means take the reciprocal.

1. Review: What exponents mean

When the exponent is a positive whole number, it tells repeated multiplication.

• \(2^3 = 2 \cdot 2 \cdot 2 = 8\)
• \(5^2 = 5 \cdot 5 = 25\)
• \(10^1 = 10\)

You may already know this rule for dividing powers with the same base:

$$\frac{a^m}{a^n} = a^{m-n} \quad \text{for } a \ne 0$$

This rule helps explain zero and negative exponents.

2. Zero exponents

Let’s look at a pattern with powers of 2:

• \(2^4 = 16\)
• \(2^3 = 8\)
• \(2^2 = 4\)
• \(2^1 = 2\)

Each time the exponent goes down by 1, the value is divided by 2.

So if we continue the pattern:

• \(2^1 = 2\)
• \(2^0 = 2 \div 2 = 1\)

This is why:

$$a^0 = 1 \quad \text{for any non-zero number } a$$

Here are some examples:

• \(7^0 = 1\)
• \((-3)^0 = 1\)
• \(100^0 = 1\)

Important: the base must be non-zero. The rule is for \(a \ne 0\).

Another way to see it is by using the division rule:

$$\frac{a^5}{a^5} = a^{5-5} = a^0$$

But any non-zero number divided by itself is 1:

$$\frac{a^5}{a^5} = 1$$

So:

$$a^0 = 1 \quad \text{when } a \ne 0$$

3. Negative exponents

Now let’s keep the same pattern going with powers of 2:

• \(2^3 = 8\)
• \(2^2 = 4\)
• \(2^1 = 2\)
• \(2^0 = 1\)

Again, each time the exponent goes down by 1, divide by 2.

• \(2^{-1} = 1 \div 2 = \frac{1}{2}\)
• \(2^{-2} = \frac{1}{2} \div 2 = \frac{1}{4}\)
• \(2^{-3} = \frac{1}{4} \div 2 = \frac{1}{8}\)

This leads to the rule:

$$a^{-n} = \frac{1}{a^n} \quad \text{for } a \ne 0$$

A negative exponent does not make the number negative. It means take the reciprocal of the positive exponent form.

For example:

• \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
• \(10^{-1} = \frac{1}{10}\)
• \(5^{-3} = \frac{1}{5^3} = \frac{1}{125}\)

4. Be careful with negatives and parentheses

Parentheses matter a lot when the base is negative.

Compare these:

• \((-2)^2 = (-2)(-2) = 4\)
• \(-2^2 = -(2^2) = -4\)

Now with a zero exponent:

• \((-2)^0 = 1\)

And with a negative exponent:

• \((-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}\)
• \((-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}\)

If the negative base is inside parentheses, include the sign as part of the base before applying the exponent.

5. Worked examples

Example 1: Evaluate a zero exponent

Find \(9^0\).

Step 1: Notice the base is 9, which is non-zero.

Step 2: Use the rule \(a^0 = 1\).

$$9^0 = 1$$

Answer: \(1\)

Example 2: Evaluate a negative exponent

Find \(4^{-2}\).

Step 1: A negative exponent means take the reciprocal.

$$4^{-2} = \frac{1}{4^2}$$

Step 2: Compute \(4^2\).

$$4^2 = 16$$

Step 3: Write the final answer.

$$4^{-2} = \frac{1}{16}$$

Answer: \(\frac{1}{16}\)

Example 3: Negative base with a negative exponent

Find \((-3)^{-3}\).

Step 1: Use the reciprocal rule.

$$(-3)^{-3} = \frac{1}{(-3)^3}$$

Step 2: Compute \((-3)^3\).

$$(-3)^3 = (-3)(-3)(-3) = 9(-3) = -27$$

Step 3: Write the answer.

$$(-3)^{-3} = \frac{1}{-27} = -\frac{1}{27}$$

Answer: \(-\frac{1}{27}\)

Example 4: Rewrite with only positive exponents

Rewrite \(\frac{1}{x^{-4}}\) using only positive exponents.

Step 1: Since \(x^{-4} = \frac{1}{x^4}\), substitute that in.

$$\frac{1}{x^{-4}} = \frac{1}{\frac{1}{x^4}}$$

Step 2: Dividing by a fraction is the same as multiplying by its reciprocal.

$$\frac{1}{\frac{1}{x^4}} = x^4$$

Answer: \(x^4\)

6. Common mistakes to avoid

• Mistake: Thinking \(a^0 = 0\).
  Correct idea: \(a^0 = 1\) for any non-zero \(a\).
• Mistake: Thinking a negative exponent makes the answer negative.
  Correct idea: a negative exponent means reciprocal, not negative.
• Mistake: Forgetting parentheses with a negative base.
  Correct idea: \((-2)^2\) and \(-2^2\) are not the same.
• Mistake: Leaving negative exponents in a final answer when asked for positive exponents only.
  Correct idea: move the factor to the other side of a fraction, or rewrite using reciprocals.

7. Quick check

• \(6^0 = 1\)
• \(2^{-3} = \frac{1}{8}\)
• \((-5)^0 = 1\)
• \((-2)^{-2} = \frac{1}{4}\)

8. Summary

Zero and negative exponents follow patterns that match the exponent rules you already know.

• For any non-zero base, \(a^0 = 1\).
• For any non-zero base, \(a^{-n} = \frac{1}{a^n}\).

Remember: a zero exponent gives 1, and a negative exponent means reciprocal. Always watch for parentheses when the base is negative.

Scientific notation is a way to write very large numbers and very small numbers in a shorter, clearer form.

In 8th grade, scientific notation means writing a number as the product of a single-digit number and a power of 10.

The general form is:

$$a \times 10^n$$

where:

• a is a number greater than or equal to 1 and less than 10.
• n is an integer exponent, which can be positive, negative, or zero.

For example, \(4.2 \times 10^5\) is in scientific notation because \(4.2\) is at least 1 but less than 10.

But \(42 \times 10^4\) is not in scientific notation, because 42 is not a single-digit number.

Why do we use scientific notation?

Some numbers are awkward to write with all their zeros. Scientific notation helps us:

• write numbers more quickly,
• read large and small numbers more easily,
• compare sizes of numbers,
• show place value clearly using powers of 10.

How to format a number in scientific notation

To write a number in scientific notation, follow these steps:

1. Find the decimal point.
2. Move the decimal so the new number is between 1 and 10.
3. Count how many places you moved the decimal.
4. Write that count as the exponent on 10.

The direction you move the decimal tells whether the exponent is positive or negative:

• If you move the decimal left, the exponent is positive.
• If you move the decimal right, the exponent is negative.

Formatting very large numbers

Very large numbers usually have a positive exponent.

When you move the decimal to the left, you are making the number part smaller, so you need a positive power of 10 to keep the value the same.

Example idea:

$$530{,}000 = 5.3 \times 10^5$$

The decimal in 530,000 starts at the end: \(530000.\)

Move it left 5 places to get \(5.3\). Since the decimal moved left 5 places, the exponent is \(5\).

Formatting very small numbers

Very small numbers usually have a negative exponent.

When you move the decimal to the right, you are making the number part larger, so you need a negative power of 10 to keep the value the same.

Example idea:

$$0.00081 = 8.1 \times 10^{-4}$$

The decimal moves right 4 places to make \(8.1\), so the exponent is \(-4\).

Important rule for correct formatting

In scientific notation, the first factor must be at least 1 and less than 10.

• Correct: \(3.7 \times 10^6\)
• Incorrect: \(37 \times 10^5\)
• Incorrect: \(0.37 \times 10^7\)

Only the first one is correctly formatted because \(3.7\) is between 1 and 10.

Worked Examples

Example 1: Write 72,000 in scientific notation.

Step 1: Put the decimal at the end: \(72000.\)

Step 2: Move the decimal left until the number is between 1 and 10.

$$72000 = 7.2$$

The decimal moved 4 places left.

Step 3: Use 4 as the exponent on 10.

$$72{,}000 = 7.2 \times 10^4$$

Example 2: Write 0.0056 in scientific notation.

Move the decimal right until the number is between 1 and 10.

$$0.0056 = 5.6$$

The decimal moved 3 places right, so the exponent is negative 3.

$$0.0056 = 5.6 \times 10^{-3}$$

Example 3: Is \(45.1 \times 10^3\) correctly formatted scientific notation?

No. The number 45.1 is greater than 10, so it is not allowed in scientific notation form.

We need to rewrite 45.1 as 4.51 and adjust the power of 10.

$$45.1 \times 10^3 = 4.51 \times 10^4$$

So the correctly formatted scientific notation is:

$$4.51 \times 10^4$$

Example 4: Write 0.00000092 in scientific notation.

Move the decimal right until the number becomes \(9.2\).

The decimal moves 7 places.

Because it moved right, the exponent is negative.

$$0.00000092 = 9.2 \times 10^{-7}$$

How to check your answer

Ask yourself these questions:

• Is the first number at least 1 and less than 10?
• Did I count the decimal moves correctly?
• Did I choose the correct sign for the exponent?

A quick check:

• If the original number was very large, the exponent should usually be positive.
• If the original number was very small, the exponent should usually be negative.

Common mistakes to avoid

• Using a first factor bigger than 10
  Example: \(23 \times 10^4\) is not correctly formatted.
• Using the wrong exponent sign
  For a small number like \(0.00034\), the exponent should be negative, not positive.
• Miscounting decimal places
  Count carefully how many places the decimal moves.
• Forgetting that whole numbers have a decimal point at the end
  For example, \(8{,}000\) can be thought of as \(8000.\)

Practice ideas

Try formatting these numbers on your own:

• \(6{,}400{,}000\)
• \(0.00045\)
• \(91{,}200\)
• \(0.0000003\)

Check that each answer has a first factor between 1 and 10.

Summary

Scientific notation writes a number as $$a \times 10^n$$ where \(a\) is at least 1 and less than 10.

To format a number, move the decimal until the first factor is between 1 and 10, then count the moves to find the exponent.

Move the decimal left for large numbers, which gives a positive exponent. Move the decimal right for small numbers, which gives a negative exponent.

Comparing Magnitudes means figuring out how many times larger one quantity is than another. In 8th Grade math, scientific notation helps us do this quickly, especially when the numbers are very large or very small.

For example, if one planet is much farther from the Sun than another, or if one cell is much smaller than a grain of sand, scientific notation makes the comparison easier to see and calculate.

In this lesson, you will learn how to compare two quantities written in scientific notation and find the factor that tells how many times greater one quantity is than the other.

Step 1: Remember scientific notation.

A number in scientific notation is written as

$$a \times 10^n$$

where:

•  is a number greater than or equal to 1 and less than 10,
• n is an integer exponent.

Examples:

• \(4.5 \times 10^6 = 4{,}500{,}000\)
• \(3.2 \times 10^{-4} = 0.00032\)

Step 2: Understand what “how many times larger” means.

To find how many times larger one number is than another, we divide.

If quantity A is larger than quantity B, then:

$$\text{How many times larger} = \frac{A}{B}$$

So if one number is 20 and another is 5, then

$$\frac{20}{5} = 4$$

This means 20 is 4 times as large as 5.

Step 3: Divide numbers in scientific notation.

When dividing scientific notation, divide the front numbers and subtract the exponents.

$$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$$

This rule is the key to comparing magnitudes.

Main idea: The power of 10 tells a lot about the size of the number. A bigger exponent usually means a much larger number.

Worked Example 1: Compare two large numbers

A stadium holds \(8 \times 10^4\) people for a special event. A school gym holds \(2 \times 10^3\) people. How many times larger is the stadium capacity than the gym capacity?

Step 1: Write the comparison as division.

$$\frac{8 \times 10^4}{2 \times 10^3}$$

Step 2: Divide the front numbers and subtract exponents.

$$\left(\frac{8}{2}\right) \times 10^{4-3} = 4 \times 10^1$$

Step 3: Simplify.

$$4 \times 10^1 = 40$$

Answer: The stadium holds 40 times as many people as the gym.

Worked Example 2: Compare two very small quantities

A bacteria cell is about \(6 \times 10^{-6}\) meters long. A virus is about \(3 \times 10^{-8}\) meters long. How many times larger is the bacteria than the virus?

Step 1: Set up the division.

$$\frac{6 \times 10^{-6}}{3 \times 10^{-8}}$$

Step 2: Divide the front numbers and subtract exponents.

$$\left(\frac{6}{3}\right) \times 10^{-6-(-8)}$$

$$2 \times 10^2$$

Step 3: Simplify.

$$2 \times 10^2 = 200$$

Answer: The bacteria is 200 times larger than the virus.

Notice something important: subtracting a negative exponent becomes addition.

$$-6 - (-8) = -6 + 8 = 2$$

Worked Example 3: Compare a large number and a smaller number

The mass of Earth is about \(6 \times 10^{24}\) kilograms. The mass of the Moon is about \(7.5 \times 10^{22}\) kilograms. About how many times greater is Earth’s mass than the Moon’s mass?

Step 1: Write the ratio.

$$\frac{6 \times 10^{24}}{7.5 \times 10^{22}}$$

Step 2: Divide coefficients and subtract exponents.

$$\left(\frac{6}{7.5}\right) \times 10^{24-22} = 0.8 \times 10^2$$

Step 3: Simplify.

$$0.8 \times 10^2 = 80$$

Answer: Earth’s mass is about 80 times greater than the Moon’s mass.

Sometimes your answer may not look like standard scientific notation at first, like \(0.8 \times 10^2\). That is okay while comparing. You can still simplify it to an ordinary number, which here is 80.

Worked Example 4: When the answer is not a whole number

One star is \(9 \times 10^5\) kilometers wide. Another is \(3.6 \times 10^5\) kilometers wide. How many times wider is the first star?

Step 1: Divide the two numbers.

$$\frac{9 \times 10^5}{3.6 \times 10^5}$$

Step 2: Divide coefficients and subtract exponents.

$$\left(\frac{9}{3.6}\right) \times 10^{5-5} = 2.5 \times 10^0$$

Step 3: Simplify.

$$2.5 \times 10^0 = 2.5$$

Answer: The first star is 2.5 times wider than the second star.

This shows that “how many times larger” does not always have to be a whole number.

Helpful strategy for comparing magnitudes

1. Decide which quantity is larger.
2. Write a division expression: larger quantity divided by smaller quantity.
3. Divide the coefficients.
4. Subtract the exponents on 10.
5. Simplify the result.
6. Write the answer in words: “___ is ___ times larger than ___.”

What the exponents tell you

If two numbers have the same coefficient, the exponent difference tells the comparison right away.

For example:

$$\frac{5 \times 10^7}{5 \times 10^3} = 1 \times 10^4 = 10{,}000$$

So \(5 \times 10^7\) is 10,000 times larger than \(5 \times 10^3\).

Each increase of 1 in the exponent means the number becomes 10 times larger. An increase of 2 means 100 times larger. An increase of 3 means 1,000 times larger.

Common mistakes to avoid

• Do not subtract the coefficients. You must divide them.
• Do not add exponents when dividing. When dividing powers of 10, you subtract exponents.
• Put the larger quantity on top if the question asks “how many times larger.”
• Be careful with negative exponents. Subtracting a negative changes to addition.

Quick Check

Try these on your own:

• How many times larger is \(4 \times 10^9\) than \(2 \times 10^7\)?
• How many times larger is \(7.2 \times 10^{-3}\) than \(9 \times 10^{-5}\)?

Answers:

• $$\frac{4 \times 10^9}{2 \times 10^7} = 2 \times 10^2 = 200$$ so the first is 200 times larger.
• $$\frac{7.2 \times 10^{-3}}{9 \times 10^{-5}} = 0.8 \times 10^2 = 80$$ so the first is 80 times larger.

Summary

Comparing magnitudes means finding how many times larger one quantity is than another. To do this with scientific notation, divide the coefficients and subtract the exponents:

$$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$$

Scientific notation makes it much easier to compare very large and very small numbers. If you follow the steps carefully, you can compare sizes in science, astronomy, technology, and everyday math problems.

Multiplying and Dividing in Scientific Notation

Scientific notation is a short way to write very large or very small numbers. A number in scientific notation is written as:

$$a \times 10^n$$

where (a") is at least 1 but less than 10, and (n") is an integer.

Examples:

• \(3.2 \times 10^5\)
• \(7.1 \times 10^{-3}\)

In this lesson, you will learn how to multiply and divide numbers written in scientific notation.

Why this works

When multiplying and dividing in scientific notation, we use exponent rules for powers of 10:

• When multiplying powers with the same base: \(10^a \cdot 10^b = 10^{a+b}\)
• When dividing powers with the same base: \(10^a \div 10^b = 10^{a-b}\)

This means we can work with the number part and the power of 10 part separately.

How to multiply numbers in scientific notation

1. Multiply the coefficients.
2. Multiply the powers of 10 by adding the exponents.
3. Check whether the answer is in proper scientific notation.
4. If the coefficient is 10 or greater, rewrite it so the coefficient is between 1 and 10.

In general:

$$ (a \times 10^m)(b \times 10^n) = (ab) \times 10^{m+n} $$

Worked Example 1: Multiply two numbers in scientific notation

Find:

$$ (2 \times 10^3)(4 \times 10^5) $$

Step 1: Multiply the coefficients.

\(2 \cdot 4 = 8\)

Step 2: Add the exponents.

\(10^3 \cdot 10^5 = 10^{3+5} = 10^8\)

Step 3: Write the answer.

$$ 8 \times 10^8 $$

This is already in scientific notation because 8 is between 1 and 10.

Worked Example 2: Multiply and then adjust the coefficient

Find:

$$ (6 \times 10^4)(3 \times 10^2) $$

Step 1: Multiply the coefficients.

\(6 \cdot 3 = 18\)

Step 2: Add the exponents.

\(10^4 \cdot 10^2 = 10^6\)

So we get:

$$ 18 \times 10^6 $$

But this is not proper scientific notation, because 18 is not between 1 and 10.

Rewrite 18 as:

$$ 18 = 1.8 \times 10^1 $$

Now combine the powers of 10:

$$ 18 \times 10^6 = (1.8 \times 10^1) \times 10^6 = 1.8 \times 10^7 $$

Final answer:

$$ 1.8 \times 10^7 $$

How to divide numbers in scientific notation

1. Divide the coefficients.
2. Divide the powers of 10 by subtracting the exponents.
3. Check whether the answer is in proper scientific notation.
4. If needed, rewrite the coefficient so it is between 1 and 10.

In general:

$$ \frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n} $$

Worked Example 3: Divide two numbers in scientific notation

Find:

$$ \frac{8 \times 10^7}{2 \times 10^3} $$

Step 1: Divide the coefficients.

\(8 \div 2 = 4\)

Step 2: Subtract the exponents.

\(10^7 \div 10^3 = 10^{7-3} = 10^4\)

Step 3: Write the answer.

$$ 4 \times 10^4 $$

This is already in scientific notation.

Worked Example 4: Divide and then adjust the coefficient

Find:

$$ \frac{3 \times 10^5}{6 \times 10^2} $$

Step 1: Divide the coefficients.

\(3 \div 6 = 0.5\)

Step 2: Subtract the exponents.

\(10^5 \div 10^2 = 10^{5-2} = 10^3\)

So we get:

$$ 0.5 \times 10^3 $$

This is not proper scientific notation, because 0.5 is less than 1.

Rewrite 0.5 as:

$$ 0.5 = 5 \times 10^{-1} $$

Now combine the powers of 10:

$$ 0.5 \times 10^3 = (5 \times 10^{-1}) \times 10^3 = 5 \times 10^2 $$

Final answer:

$$ 5 \times 10^2 $$

A quick way to check your answer

• The coefficient should be at least 1 and less than 10.
• If you multiply numbers greater than 1, the product should usually be larger.
• If you divide by a large number, the result should get smaller.
• Make sure you added exponents when multiplying and subtracted exponents when dividing.

Common mistakes to avoid

• Adding exponents when dividing. For division, you must subtract.
• Forgetting to adjust the coefficient. Answers like \(15 \times 10^4\) or \(0.3 \times 10^6\) are not in proper scientific notation.
• Mixing up coefficient and exponent work. Multiply or divide the coefficients normally, and use exponent rules only for the powers of 10.

Step-by-step strategy

When you see a multiplication or division problem in scientific notation, ask yourself:

1. What do I do with the coefficients?
2. What do I do with the exponents?
3. Is my final coefficient between 1 and 10?

Practice thinking

Try these in your head:

• \((5 \times 10^2)(2 \times 10^3)\) gives coefficient \(5 \cdot 2 = 10\), so you will need to rewrite.
• \((9 \times 10^6) \div (3 \times 10^2)\) gives coefficient \(9 \div 3 = 3\), so it is likely already in proper form.

Summary

To multiply numbers in scientific notation, multiply the coefficients and add the exponents of 10.

To divide numbers in scientific notation, divide the coefficients and subtract the exponents of 10.

Always make sure your final answer is in proper scientific notation, with a coefficient that is at least 1 but less than 10.

Adding and Subtracting in Scientific Notation

Scientific notation is a way to write very large or very small numbers using powers of 10. A number in scientific notation is written like this:

Here, the number 07a is the coefficient, and it must be at least 1 but less than 10. The exponent n tells how many places the decimal point has been moved.

For example:

• $$3.2 \times 10^4 = 32{,}000$$
• $$5.6 \times 10^{-3} = 0.0056$$

Multiplying and dividing in scientific notation can be done by working with the coefficients and exponents separately. But adding and subtracting are different.

When you add or subtract numbers in scientific notation, the powers of 10 must match first. This is similar to adding like terms in algebra: you can add the coefficients only when the exponential parts are the same.

For example, you can add:

because both numbers have the same power of 10.

But you cannot directly add:

because the exponents are different. First, you must rewrite one of the numbers so both use the same power of 10.

Step-by-step method

1. Check whether the powers of 10 are the same.
2. If they are different, rewrite one number so both have the same exponent.
3. Add or subtract the coefficients.
4. Make sure the final answer is in proper scientific notation.

How to rewrite a number with a different exponent

Changing the exponent by 1 changes the coefficient by a factor of 10 in the opposite direction.

• $$3.5 \times 10^6 = 35 \times 10^5$$
• $$35 \times 10^5 = 3.5 \times 10^6$$

This means you can rewrite a number to match another exponent without changing its value.

Worked Example 1: Same exponents

Add:

The exponents are already the same, so add the coefficients:

Keep the power of 10 the same:

Answer: $$7.5 \times 10^3$$

Worked Example 2: Different exponents

Add:

The exponents are different: 4 and 3. Rewrite one number so both use the same exponent.

Lets rewrite $$3.2 \times 10^4$$ using $$10^3$$:

Now the expression is:

Add the coefficients:

So we get:

This is not yet in proper scientific notation because 36.5 is greater than 10. Rewrite it:

Answer: $$3.65 \times 10^4$$

Worked Example 3: Subtracting with small numbers

Subtract:

The exponents are different: 2 and 3. Rewrite one number so both have the same exponent.

Rewrite $$7.8 \times 10^{-2}$$ using $$10^{-3}$$:

Now subtract:

Rewrite in proper scientific notation:

Answer: $$7.57 \times 10^{-2}$$

Worked Example 4: Subtracting when the result stays with the same exponent

Subtract:

The exponents match, so subtract the coefficients:

Keep the power of 10:

Answer: $$5.7 \times 10^6$$

Important idea to remember

When adding or subtracting in scientific notation, you are really combining numbers with the same place value size. The exponent tells that size. If the exponents do not match, the numbers are not in the same-sized groups yet.

Common mistakes to avoid

• Do not add the exponents. That is for multiplication, not addition.
• Do not add the coefficients unless the exponents match first.
• Always check the final answer. The coefficient must be at least 1 and less than 10.
• Be careful with negative exponents. A number like $$10^{-2}$$ is larger than $$10^{-3}$$ because $$0.01 > 0.001$$.

Quick check

Try to think through these:

• $$\left(6 \times 10^5\right) + \left(2 \times 10^5\right) = 8 \times 10^5$$
• $$\left(4 \times 10^2\right) + \left(3 \times 10^1\right)$$ must be rewritten first because the exponents are different.

Summary

To add or subtract in scientific notation, first make the powers of 10 the same. Then combine the coefficients and keep the common power of 10. Finally, rewrite the result so the coefficient is at least 1 but less than 10.

If you remember match exponents first, you will be able to solve these problems correctly.

Technological Representations of Exponents means understanding how calculators, computers, and apps show very large and very small numbers.

When numbers are too long to fit nicely on a screen, technology often uses E-notation. This is another way to write numbers in scientific notation.

If you already know scientific notation, then E-notation is the same idea with a different symbol. The letter E stands for “times ten to the power of.”

For example, the scientific notation $$3.2 \times 10^5$$ may appear on a calculator as $$3.2\text{E}5$$ or sometimes as 3.2e5 in software.

This lesson will help you read E-notation, write it as scientific notation, and understand what the exponent means.

1. Review: Scientific Notation

Scientific notation is a short way to write very large or very small numbers.

• A number in scientific notation has the form $$a \times 10^n$$
• The number $$a$$ must be at least 1 and less than 10.
• The exponent $$n$$ tells how many places the decimal moves.

Examples:

• $$4.7 \times 10^3 = 4700$$
• $$6.1 \times 10^{-2} = 0.061$$

2. What E-Notation Means

On a calculator or computer, the expression $$\times 10^n$$ is often replaced by E.

So these mean the same thing:

• $$5.4 \times 10^6$$
• $$5.4\text{E}6$$

And these also mean the same thing:

• $$8.2 \times 10^{-4}$$
• $$8.2\text{E}-4$$

You can think of E-notation like this:

$$a\text{E}n = a \times 10^n$$

3. How to Read Positive Exponents in E-Notation

If the exponent after the E is positive, the number is large. Move the decimal point to the right.

Example:

$$2.8\text{E}4 = 2.8 \times 10^4$$

Since the exponent is 4, move the decimal 4 places to the right:

$$2.8 \to 28 \to 280 \to 2800 \to 28000$$

So, $$2.8\text{E}4 = 28000$$.

4. How to Read Negative Exponents in E-Notation

If the exponent after the E is negative, the number is small. Move the decimal point to the left.

Example:

$$7.1\text{E}-3 = 7.1 \times 10^{-3}$$

Since the exponent is -3, move the decimal 3 places to the left:

$$7.1 \to 0.71 \to 0.071 \to 0.0071$$

So, $$7.1\text{E}-3 = 0.0071$$.

5. How to Change Scientific Notation to E-Notation

This is usually very simple. Replace $$\times 10^n$$ with E followed by the exponent.

• $$9.3 \times 10^7 = 9.3\text{E}7$$
• $$4.5 \times 10^{-6} = 4.5\text{E}-6$$

6. How to Change E-Notation to Scientific Notation

Replace the E with $$\times 10^{}$$ and keep the exponent.

• $$6.8\text{E}3 = 6.8 \times 10^3$$
• $$1.2\text{E}-5 = 1.2 \times 10^{-5}$$

7. Why Technology Uses E-Notation

Calculators and computers have limited screen space. Writing a number like $$0.00000034$$ or $$560000000$$ can take up a lot of room.

E-notation is faster, shorter, and easier for technology to display. It also helps avoid mistakes when reading long strings of zeros.

For example:

• $$0.00000034 = 3.4 \times 10^{-7} = 3.4\text{E}-7$$
• $$560000000 = 5.6 \times 10^8 = 5.6\text{E}8$$

8. Worked Examples

Example 1: Read a calculator display

A calculator shows 4.2E3. Write this in scientific notation and standard form.

Step 1: Replace E with $$\times 10^{}$$.

$$4.2\text{E}3 = 4.2 \times 10^3$$

Step 2: The exponent is positive 3, so move the decimal 3 places to the right.

$$4.2 \to 42 \to 420 \to 4200$$

Answer: Scientific notation: $$4.2 \times 10^3$$

Standard form: $$4200$$

Example 2: Read a small number from software

A computer displays 6.5e-4. Write it in scientific notation and standard form.

Step 1: The lowercase e means the same as E.

$$6.5\text{e}-4 = 6.5 \times 10^{-4}$$

Step 2: The exponent is negative 4, so move the decimal 4 places to the left.

$$6.5 \to 0.65 \to 0.065 \to 0.0065 \to 0.00065$$

Answer: Scientific notation: $$6.5 \times 10^{-4}$$

Standard form: $$0.00065$$

Example 3: Change scientific notation to technological form

Write $$3.09 \times 10^6$$ in E-notation.

Step 1: Keep the first number, 3.09.

Step 2: Replace $$\times 10^6$$ with E6.

$$3.09 \times 10^6 = 3.09\text{E}6$$

Answer: 3.09E6

Example 4: Interpret a real technology display

A calculator gives the answer 9.81E-2. What number is this?

Step 1: Write it as scientific notation.

$$9.81\text{E}-2 = 9.81 \times 10^{-2}$$

Step 2: Move the decimal 2 places to the left.

$$9.81 \to 0.981 \to 0.0981$$

Answer: $$0.0981$$

9. Common Mistakes to Avoid

• Forgetting what E means: E does not mean “add.” It means $$\times 10^n$$.
• Moving the decimal the wrong direction: Positive exponent means right. Negative exponent means left.
• Ignoring the sign of the exponent: $$2.4\text{E}5$$ and $$2.4\text{E}-5$$ are very different numbers.
• Misreading lowercase e: In many apps, e and E mean the same thing.

10. Quick Check

1. Write $$7.3\text{E}2$$ in standard form.
2. Write $$5.9\text{E}-3$$ in scientific notation.
3. Write $$8.1 \times 10^4$$ in E-notation.
4. Which is larger: $$3.2\text{E}5$$ or $$3.2\text{E}-5$$?

Answers:

1. $$730$$
2. $$5.9 \times 10^{-3}$$
3. $$8.1\text{E}4$$
4. $$3.2\text{E}5$$

Summary

E-notation is the way technology writes scientific notation. The expression $$a\text{E}n$$ means $$a \times 10^n$$.

If the exponent is positive, move the decimal to the right. If the exponent is negative, move the decimal to the left.

When you see E-notation on a calculator or computer, you can translate it into scientific notation and then into standard form. This helps you understand large and small numbers correctly.