Exponent Rules: Product, Quotient and Power

Exponent rules are shortcuts for simplifying powers. When the bases match, multiply by adding the exponents: $x^a \cdot x^b = x^{a+b}$ . Divide by subtracting them. For a power of a power, multiply the exponents. For a power of a product or quotient, give the exponent to each factor.

Before this lesson: You should know what exponents mean, how to work with algebraic expressions, and how to do combining like terms.

Extra practice: The Algebra 1 Unit 6 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

Why the Exponent Rules Work

An exponent counts how many times a base is multiplied. So $x^3$ means $x \cdot x \cdot x$ . The base is $x$ and the exponent is 3.

Now multiply $x^3 \cdot x^2$ . Write out every factor: $(x \cdot x \cdot x)(x \cdot x)$ . That is 5 factors of $x$ , so the answer is $x^5$ . You added 3 and 2.

Raise $x^2$ to the 3rd power and you get $x^2 \cdot x^2 \cdot x^2$ . That is 3 groups of 2, or 6 factors. So $(x^2)^3 = x^6$ . You multiplied 2 and 3.

Two rows. Top row: x³ · x² is drawn as 3 blue boxes labeled x and 2 orange boxes labeled x, which together make 5 boxes, so x³ · x² = x⁵ (add the exponents). Bottom row: (x²)³ is drawn as 3 groups of 2 boxes, 6 boxes in all, so (x²)³ = x⁶ (multiply the exponents). — Count the factors of x. Multiplying powers adds the exponents. A power of a power multiplies them.

The Exponent Rules

In each rule, $a$ and $b$ are positive whole numbers. The bases can’t be 0 when they are on the bottom of a fraction.

Rule	In symbols	Example
Product rule	$x^a \cdot x^b = x^{a+b}$	$x^4 \cdot x^3 = x^7$
Quotient rule	$\dfrac{x^a}{x^b} = x^{a-b}$	$\dfrac{x^9}{x^4} = x^5$
Power of a power	$(x^a)^b = x^{ab}$	$(x^5)^2 = x^{10}$
Power of a product	$(xy)^a = x^a y^a$	$(3x)^2 = 9x^2$
Power of a quotient	$\left(\dfrac{x}{y}\right)^a = \dfrac{x^a}{y^a}$	$\left(\dfrac{x}{2}\right)^3 = \dfrac{x^3}{8}$

How to Simplify with Exponent Rules

Multiply or divide the numbers (the coefficients) first.
Group the same bases together.
Use the rules on each base: add exponents to multiply, subtract to divide, multiply for a power of a power.
If a whole product is raised to a power, raise every factor, including the number.
Write each base once, in alphabetical order.

In this lesson, the quotient rule leaves a positive exponent. What $x^0$ and negative exponents mean is the next lesson, Zero and Negative Exponents.

Worked Examples

Example 1: Product Rule

Simplify (a) $2^3 \cdot 2^4$ and (b) $x^5 \cdot x \cdot x^2$ .

2^3 \cdot 2^4 = 2^{3+4}

Step 1. (a) The bases match, so keep the base 2 and add the exponents. Do not multiply the bases. (See exponents.)

2^{3+4} = 2^7

Step 2. Add the exponents.

2^7 = 128

Step 3. Find the value: $2^7$ is seven 2s multiplied together.

x^5 \cdot x \cdot x^2 = x^5 \cdot x^1 \cdot x^2

Step 4. (b) A plain $x$ has exponent 1. Write it in.

x^{5+1+2} = x^8

Step 5. The bases match, so add the exponents.

Answer: (a) $2^7 = 128$ ; (b) $x^8$

Example 2: Monomials with Coefficients

Simplify $(3x^4y)(-5x^2y^3)$ .

3 \cdot (-5) \cdot x^4 \cdot x^2 \cdot y \cdot y^3

Step 1. Group the numbers and the like bases together.

3 \cdot (-5) = -15

Step 2. Multiply the numbers. A positive times a negative is negative (see multiplying and dividing integers).

x^4 \cdot x^2 = x^{4+2} = x^6

Step 3. Add the exponents on $x$ .

y^1 \cdot y^3 = y^{1+3} = y^4

Step 4. Add the exponents on $y$ . A plain $y$ has exponent 1.

Answer: $-15x^6y^4$

Example 3: Quotient Rule

Simplify $\dfrac{24a^7b^3}{6a^2b}$ .

\frac{24}{6} = 4

Step 1. Divide the numbers.

\frac{a^7}{a^2} = a^{7-2} = a^5

Step 2. Subtract the exponents on $a$ : top exponent minus bottom exponent.

\frac{b^3}{b^1} = b^{3-1} = b^2

Step 3. Subtract the exponents on $b$ . A plain $b$ has exponent 1.

Answer: $4a^5b^2$

Example 4: Power Rules

Simplify (a) $(2x^3)^4$ , (b) $(-3y^2)^3$ and (c) $\left(\dfrac{x^2}{5}\right)^3$ .

(2x^3)^4 = 2^4 \cdot (x^3)^4

Step 1. (a) Raise each factor to the 4th power.

16 \cdot x^{3 \cdot 4} = 16x^{12}

Step 2. Find $2^4 = 16$ . For a power of a power, multiply the exponents.

(-3y^2)^3 = (-3)^3 \cdot (y^2)^3

Step 3. (b) Raise each factor to the 3rd power.

(-3)^3 = (-3)(-3)(-3) = -27

Step 4. Find the number. An odd power of a negative number is negative.

-27 \cdot y^{2 \cdot 3} = -27y^6

Step 5. Multiply the exponents on y.

\left(\frac{x^2}{5}\right)^3 = \frac{(x^2)^3}{5^3}

Step 6. (c) Raise the top and the bottom to the 3rd power.

\frac{x^6}{125}

Step 7. Multiply the exponents on top, and find $5^3 = 125$ .

Answer: (a) $16x^{12}$ ; (b) $-27y^6$ ; (c) $\dfrac{x^6}{125}$

Example 5: Word Problem

A cube has edges that are $3x^2$ centimeters long. Write its volume as a single term. Then find the volume when $x = 2$ .

V = (3x^2)^3

Step 1. The volume of a cube is the edge cubed.

V = 3^3 \cdot (x^2)^3

Step 2. Use the power of a product rule. Raise each factor to the 3rd power.

V = 27x^6

Step 3. Find $3^3 = 27$ and multiply the exponents on x.

V = 27 \cdot 2^6

Step 4. Substitute $x = 2$ .

27 \cdot 64 = 1728

Step 5. Find $2^6 = 64$ and multiply.

Answer: The volume is $27x^6$ cubic centimeters. When $x = 2$ , it is 1,728 cubic centimeters.

Check: The edge is $3 \cdot 2^2 = 3 \cdot 4 = 12$ cm, and $12^3 = 1728$ .

Common Mistakes

Mistake	Example of the error	Fix
Multiplying exponents when multiplying powers	$x^3 \cdot x^4 = x^{12}$	Add them: $x^3 \cdot x^4 = x^7$
Multiplying the bases	$2^3 \cdot 2^4 = 4^7$	Keep the base: $2^3 \cdot 2^4 = 2^7$
Combining different bases	$x^2 \cdot y^3 = (xy)^5$	$x^2y^3$ can’t be simplified
Not raising the number	$(2x^3)^4 = 2x^{12}$	Raise every factor: $16x^{12}$
Adding exponents when adding terms	$x^2 + x^3 = x^5$	These are not like terms. $x^2 + x^3$ stays as it is
Using the product rule on a sum	$(x + y)^2 = x^2 + y^2$	The rule is only for products: $(xy)^2 = x^2y^2$

Checking Your Work with Numbers

Not sure a step is right? Put in a small number and compare both sides. Try $x = 2$ .

Step to test	Left side	Right side	Match?
$x^3 \cdot x^4 = x^7$	$8 \cdot 16 = 128$	$2^7 = 128$	Yes
$x^3 \cdot x^4 = x^{12}$	$128$	$2^{12} = 4096$	No, so this step is wrong
$(x^2)^3 = x^6$	$4^3 = 64$	$2^6 = 64$	Yes

Exponent rules are used all the time with powers of 10. In scientific notation, $(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$ , because $10^4 \cdot 10^5 = 10^9$ . You will also use them to multiply polynomials in Unit 7.

Practice Problems

Simplify $x^6 \cdot x^3$ .
Simplify $(4a^2)(7a^5)$ .
Simplify $\dfrac{y^{10}}{y^4}$ .
Simplify $(m^3)^5$ .
Simplify $(-2x^4)^3$ .
Simplify $\dfrac{18x^8y^5}{3x^3y^2}$ .
Simplify $\left(\dfrac{a}{3}\right)^4$ .
Simplify $(2x^2y)^3(xy^2)$ .
Find the value of $\dfrac{5^9}{5^7}$ without a calculator.
A square tile has sides $4n^3$ inches long. Write its area as a single term.

Show answers

x^6 \cdot x^3 = x^{6+3}

The bases match, so add the exponents.

x^9

Add.

4 \cdot 7 = 28

Multiply the numbers.

a^2 \cdot a^5 = a^{2+5} = a^7

Add the exponents on a.

So the answer is $28a^7$ .

\frac{y^{10}}{y^4} = y^{10-4}

Subtract the exponents.

y^6

Subtract.

(m^3)^5 = m^{3 \cdot 5}

For a power of a power, multiply the exponents.

m^{15}

Multiply.

(-2x^4)^3 = (-2)^3 \cdot (x^4)^3

Raise each factor to the 3rd power.

(-2)^3 = (-2)(-2)(-2) = -8

Find the number. An odd power of a negative is negative.

(x^4)^3 = x^{12}

Multiply the exponents on x.

So the answer is $-8x^{12}$ .

\frac{18}{3} = 6

Divide the numbers.

\frac{x^8}{x^3} = x^{8-3} = x^5

Subtract the exponents on x.

\frac{y^5}{y^2} = y^{5-2} = y^3

Subtract the exponents on y.

So the answer is $6x^5y^3$ .

\left(\frac{a}{3}\right)^4 = \frac{a^4}{3^4}

Raise the top and the bottom to the 4th power.

3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81

Find the bottom.

So the answer is $\dfrac{a^4}{81}$ .

(2x^2y)^3 = 2^3 \cdot (x^2)^3 \cdot y^3

Raise each factor in the first part to the 3rd power.

8x^6y^3

Simplify.

8x^6y^3 \cdot xy^2 = 8x^{6+1}y^{3+2}

Multiply by $xy^2$ . Add the exponents on each base.

8x^7y^5

Add.

\frac{5^9}{5^7} = 5^{9-7}

The bases match, so subtract the exponents.

5^2

Subtract.

5^2 = 25

Find the value.

10.

A = (4n^3)^2

The area of a square is the side squared.

A = 4^2 \cdot (n^3)^2

Raise each factor to the 2nd power.

A = 16n^6

Find $4^2 = 16$ and multiply the exponents.

The area is $16n^6$ square inches.

Want more practice? The Algebra 1 Unit 6 Practice Pack has 49 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

FAQ

What are the exponent rules?

The main rules are the product rule ( $x^a \cdot x^b = x^{a+b}$ ), the quotient rule ( $\frac{x^a}{x^b} = x^{a-b}$ ) and the power rules. The power rules say $(x^a)^b = x^{ab}$ and $(xy)^a = x^a y^a$ .

When do you add exponents and when do you multiply them?

Add them when you multiply two powers with the same base, like $x^2 \cdot x^3 = x^5$ . Multiply them when you raise a power to a power, like $(x^2)^3 = x^6$ .

Can you use exponent rules with different bases?

Not the product and quotient rules. $x^2 \cdot y^3$ stays as it is. But if the exponents match, you can group them: $x^3y^3 = (xy)^3$ .

Is (x + y)² the same as x² + y²?

No. The power rule works for products, not sums. For example, $(2 + 3)^2 = 25$ , but $2^2 + 3^2 = 13$ .

Summary

Product rule: same base, add the exponents. $x^a \cdot x^b = x^{a+b}$
Quotient rule: same base, subtract the exponents. $\frac{x^a}{x^b} = x^{a-b}$
Power of a power: multiply the exponents. $(x^a)^b = x^{ab}$
Power of a product or quotient: raise every factor, including numbers.
Numbers multiply or divide as usual. Different bases can’t be combined.

Previous lesson: Systems of Inequalities

Next lesson: Zero and Negative Exponents

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