CCSS Math: 8.EE.A.1
Review the common properties of exponents that allow us to rewrite powers in different ways. For example, x²⋅x³ can be written as x⁵.
PropertyExample
xnxm=xn+mx^n\cdot x^m=x^{n+m}2325=282^3\cdot 2^5=2^8
xnxm=xnm\dfrac{x^n}{x^m}=x^{n-m}3832=36\dfrac{3^8}{3^2}=3^6
(xn)m=xnm\left(x^n\right)^m=x^{n\cdot m}(54)3=512\left(5^4\right)^3=5^{12}
(xy)n=xnyn(x\cdot y)^n=x^n\cdot y^n(35)7=3757(3\cdot 5)^7=3^7\cdot 5^7
(xy)n=xnyn\left(\dfrac xy\right)^n=\dfrac{x^n}{y^n}(23)5=2535\left(\dfrac23\right)^5=\dfrac{2^5}{3^5}
Want to learn more about these properties? Check out this video and this video.

Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.
xnxm=xn+mx^n\cdot x^m=x^{n+m}

Example

5255=52+5=575^2\cdot 5^5=5^{2+5}=5^7
5255=552 times555555 times=55555557 times=57\begin{aligned} 5^2\cdot 5^5&=\underbrace{5\cdot 5}_\text{2 times}\cdot\underbrace{5\cdot 5\cdot 5\cdot 5\cdot 5}_\text{5 times} \\\\\\ &=\underbrace{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}_\text{7 times} \\\\ &=5^7 \end{aligned}

Practice

Problem 1.1
Simplify.
Rewrite the expression in the form 8n8^n.
8684=8^6\cdot 8^4=

Want to try more problems like these? Check out this exercise.

Quotient of powers

This property states that when dividing two powers with the same base, we subtract the exponents.
xnxm=xnm\dfrac{x^n}{x^m}=x^{n-m}

Example

3832=382=36\dfrac{3^8}{3^2}=3^{8-2}=3^6
3832=333333338 times332 times=3333336 times=36\begin{aligned} \dfrac{3^8}{3^2}&=\dfrac{\overbrace{\cancel 3\cdot \cancel 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3}^\text{8 times}}{\underbrace{\cancel 3\cdot \cancel 3}_\text{2 times}} \\\\\\ &=\underbrace{3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3}_\text{6 times} \\\\ &=3^6 \end{aligned}

Practice

Problem 2.1
Simplify.
Rewrite the expression in the form 7n7^n.
7773=\dfrac{7^{7}}{7^3}=

Want to try more problems like these? Check out this exercise.

Power of a power property

This property states that to find a power of a power we multiply the exponents.
(xn)m=xnm\left(x^n\right)^m=x^{n\cdot m}

Example

(82)3=823=86\left(8^2\right)^3=8^{2\cdot3}=8^6
(82)3=8282823 times=882 times882 times882 times3 times=86\begin{aligned} \left(8^2\right)^{3}&=\underbrace{8^2\cdot 8^2\cdot 8^2}_\text{3 times} \\\\\\ &=\underbrace{ \underbrace{8\cdot 8}_\text{2 times} \cdot \underbrace{8\cdot 8}_\text{2 times} \cdot \underbrace{8\cdot 8}_\text{2 times}} _\text{3 times} \\\\ &=8^6 \end{aligned}

Practice

Problem 3.1
Simplify.
Rewrite the expression in the form 2n2^n.
(24)2=\left(2^4\right)^{2}=

Want to try more problems like these? Check out this exercise.

Power of a product

This property states that when taking the power of a product, we multiply the powers of the factors.
(xy)n=xnyn(x\cdot y)^n=x^n\cdot y^n

Example

(35)6=3656(3\cdot 5)^6=3^6\cdot 5^6
(35)6=(35)(35)(35)(35)(35)(35)6 times=3333336 times5555556 times=3656\begin{aligned} (3\cdot 5)^6&=\underbrace{(3\cdot 5)(3\cdot 5)(3\cdot 5)(3\cdot 5)(3\cdot 5)(3\cdot 5)}_\text{6 times} \\\\\\ &=\underbrace{3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3}_\text{6 times}\cdot\underbrace{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}_\text{6 times} \\\\ &=3^6\cdot 5^6 \end{aligned}

Practice

Problem 4.1
Select the equivalent expression.
(47)8=?(4\cdot 7)^8=?
Choose 1 answer:
Choose 1 answer:

Want to try more problems like these? Check out this exercise.

Power of a quotient

This property states that when taking the power of a quotient, we divide the powers of the numerator and of the denominator.
(xy)n=xnyn\left(\dfrac xy\right)^n=\dfrac{x^n}{y^n}

Example

(72)8=7828\left(\dfrac72\right)^8=\dfrac{7^8}{2^8}
(72)8=(72)(72)(72)(72)(72)(72)(72)(72)8 times=777777778 times222222228 times=7828\begin{aligned} \left(\dfrac72\right)^8&=\underbrace{\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)\left(\dfrac72\right)}_\text{8 times} \\\\\\ &=\dfrac{\overbrace{7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7}^\text{8 times}}{\underbrace{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2}_\text{8 times}} \\\\ &=\dfrac{7^8}{2^8} \end{aligned}

Practice

Problem 5.1
Select the equivalent expression.
(65)9=?\left(\dfrac65\right)^9=?
Choose 1 answer:
Choose 1 answer:

Want to try more problems like these? Check out this exercise.