Powers of Products and Quotients | Grade 8 Math | LevelKite

In $5^3 = 5 \cdot 5 \cdot 5$ , the base 5 is used as a factor (a number being multiplied), and the exponent 3 counts how many times. Every exponent in this lesson is a positive whole number ( $1, 2, 3, \dots$ ).

You already know how to raise a power to a power, as in $(5^2)^3 = 5^6$ . Now you will raise a product (the result of multiplying, such as $2 \cdot 5$ ) or a quotient (the result of dividing, such as $\frac{2}{3}$ ) to a power.

A power of a product is a product raised to an exponent, such as $(2 \cdot 5)^3$ .
A power of a quotient is a quotient raised to an exponent, such as $\left(\frac{2}{3}\right)^4$ .

Start with what a power of a product means. In $(2 \cdot 5)^3$ , the parentheses make the whole product $2 \cdot 5$ the base, so it is used as a factor 3 times. You can then reorder and regroup the factors, because multiplication gives the same result in any order and with any grouping (the commutative and associative properties):

(2 \cdot 5)^3 = (2 \cdot 5)(2 \cdot 5)(2 \cdot 5) = (2 \cdot 2 \cdot 2)(5 \cdot 5 \cdot 5) = 2^3 \cdot 5^3

Each copy of the base holds one 2 and one 5, so each of those factors ends up used 3 times. That regrouping is the reason the two forms are equal, and a quick calculation confirms it: $(2 \cdot 5)^3 = 10^3 = 1000$ , and $2^3 \cdot 5^3 = 8 \cdot 125 = 1000$ .

A power of a quotient works the same way. When you multiply fractions, you multiply the numerators and multiply the denominators. So in $\left(\frac{2}{3}\right)^4$ , the four 2s gather in the numerator and the four 3s gather in the denominator:

\left(\frac{2}{3}\right)^4 = \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{2 \cdot 2 \cdot 2 \cdot 2}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{2^4}{3^4} = \frac{16}{81}

Key concept

Power of a product and power of a quotient

To raise a product to a power, raise every factor to that power. To raise a quotient to a power, raise both the numerator and the denominator to that power. Here $a$ , $b$ and $c$ are real numbers and $n$ is a positive whole number. In the quotient, $b \neq 0$ , because a fraction with denominator 0 is undefined.

A product of three or more factors works the same way: $(abc)^n = a^n b^n c^n$ .
The parentheses decide the base. In $(-a)^n$ the base is $-a$ , but in $-a^n$ the exponent applies only to $a$ .
The rule is for multiplication and division only, not for a sum or difference such as $(a + b)^n$ .

(ab)^n = a^n b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \ b \neq 0

Worked example

Volume of a cube

A cube has edges that are $4y$ centimeters long, where $y$ is a positive number. Write an expression for its volume, and simplify it.

Let $V$ be the volume in cubic centimeters. The volume of a cube is its edge length used as a factor 3 times, so cube the edge length. Put $4y$ in parentheses so that the whole edge length is the base.
$V = (4y)^3$
The base $4y$ is the product of the factors $4$ and $y$ , so the power of a product lets you raise each factor to the third power.
$(4y)^3 = 4^3 \cdot y^3$
Evaluate $4^3$ : the exponent 3 means three factors of 4.
$4^3 \cdot y^3 = (4 \cdot 4 \cdot 4) \cdot y^3 = 64y^3$

Answer

The cube's volume is $64y^3$ cubic centimeters, which is 64 times the volume of a cube with edges $y$ centimeters long.

Worked example

A negative factor

Simplify $(-2m)^4$ , where $m$ is any real number.

The parentheses make all of $-2m$ the base. Its factors are $-2$ and $m$ , so raise each factor to the fourth power. Keep $-2$ in parentheses, because the negative sign is part of that factor.
$(-2m)^4 = (-2)^4 \cdot m^4$
Multiply four factors of $-2$ , one pair at a time. Each pair of negative factors gives a positive product, so the result is positive. Without the parentheses, $-2^4$ would mean $-(2^4) = -16$ , a different number.
$(-2)^4 = (-2)(-2) \cdot (-2)(-2) = 4 \cdot 4 = 16$
Replace $(-2)^4$ with its value from the previous step.
$(-2)^4 \cdot m^4 = 16m^4$

Answer

For any real number $m$ , $(-2m)^4 = 16m^4$ .

Worked example

A quotient with powers inside

Simplify $\left(\frac{2x^3}{y^2}\right)^4$ , where $x$ is any real number and $y \neq 0$ .

Use the power of a quotient. It applies because $y \neq 0$ , so the denominator $y^2 = y \cdot y$ is not 0 and the quotient is defined.
$\left(\frac{2x^3}{y^2}\right)^4 = \frac{(2x^3)^4}{(y^2)^4}$
Use the power of a product on the numerator, whose factors are $2$ and $x^3$ .
$\frac{(2x^3)^4}{(y^2)^4} = \frac{2^4 \cdot (x^3)^4}{(y^2)^4}$
Evaluate $2^4$ . For $(x^3)^4$ and $(y^2)^4$ , use the power of a power, which multiplies the exponents.
$2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16, \quad (x^3)^4 = x^{3 \cdot 4} = x^{12}, \quad (y^2)^4 = y^{2 \cdot 4} = y^8$
Substitute these values into the fraction from the previous step.
$\left(\frac{2x^3}{y^2}\right)^4 = \frac{16x^{12}}{y^8}$

Answer

For any real number $x$ and any real number $y \neq 0$ , $\left(\frac{2x^3}{y^2}\right)^4 = \frac{16x^{12}}{y^8}$ .

Takeaway: when parentheses hold a product or a quotient, an exponent outside them applies to each part that is multiplied or divided inside. For a positive whole number $n$ , $(ab)^n = a^n b^n$ , and $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ when $b \neq 0$ . Raise the numbers as well as the variables, and keep a negative factor such as $-2$ in parentheses when you raise it. Next, you will use these properties to evaluate expressions and to identify equivalent expressions involving exponents.