Power of a Power Rule

Power of a Power Rule

Learning Objective: Students will be able to rewrite expressions involving rational exponents using the properties of exponents.

Power to a Power rule

​The power to a power rule is useful when you have situations such as (x²)³. The exponent outside the parenthesis tells you to multiply everything inside times itself a certain number of times.

​Ex: (x²)³= x²⋅x²⋅x²= x⁶

​

Ex: (ab²)²= ab²⋅​ab²

or a⋅a⋅b⋅b⋅b⋅b = a²b⁴

Multiple Choice

Which shows the Power Rule used correctly?  $$\left(ab\right)^5$$  

• $$ab^5$$  

• $$a^5b^5$$  

• $$a^5b$$  

• $$a^6b^6$$  

Multiple Choice

Simplify the following expression:

(x⁵)⁴

• x⁹

• x²⁰

• x

• x⁵⁴

Multiple Choice

Simplify the following expression:

(b³)⁸

• b¹¹

• 8b³

• b²⁴

• 24b

Multiple Choice

Simplify. $$\left(m^3\right)^5$$  

• $$m^8$$  

• $$m^{15}$$  

• $$8m$$  

• $$15m$$  

Multiple Choice

$$\left(x^5y^7\right)^3\ =\ x^{15}y^{21}$$  

• True

• False

Multiple Choice

Simplify the following expression:

(3y⁴)²

• 9y⁸

• 9y⁶

• 6y⁸

• 6y⁶

POWER RULE

EXAMPLES

(a³)² = (a ⋅ a ⋅ a)² = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a⁶

​

​​SHORTCUT: just MULTIPLY the EXPONENTS

(w⁷)⁴​ =

​(a^x)^y = a^x⋅y

Just like before the variable terms may have a coefficient

​Ex.: (5x⁷)²= 5x⁷⋅5x⁷=25x¹⁴

Multiple Choice

$$\left(3n^4\right)^3$$  

• $$9n^{12}$$  

• $$9n^7$$  

• $$27n^7$$  

• $$27n^{12}$$  

Multiple Choice

(2ab)³

• 2³a³b³
• 2ab³
• 6a³b³
• 6ab

Multiple Choice

Simplify the following expression:

(3x³y⁵)⁴

• 3x¹²y²⁰

• 81x¹²y²⁰

• 12x¹²y²⁰

• 81x⁷y⁹

Multiple Choice

Simplify. $$\left(2cd^2\right)^4$$  

• $$2^4c^4d^6$$  

• $$2^4c^4d^8$$  

Multiple Choice

Simplify the exponential expression. $$\left(\left(-2\right)^5\right)^8$$  

• $$\left(-2\right)^{40}$$  

• $$\left(-2\right)^{13}$$  

• $$\left(-2\right)^3$$  

• $$\left(-2\right)^{16}$$  

Multiple Choice

Simplify the exponential expression. $$\left(3^2a^4\right)^5$$  

• $$3^{10}a^9$$  

• $$3^{10}a^{20}$$  

• $$3^7a^9$$  

• $$3^7a^{20}$$  

​

​(a³b⁶c^-2)² = a^3⋅2 ⋅ b^6⋅2⋅ c^-2⋅2

POWER RULE

​=

With Coefficients that are not 1

​(4x³y⁴)² = (4¹x³y⁴)²

​=

With multiple variables

​(a³b⁶c^-2)⁰ =

POWER RULE

​=

With exponents that are ZERO

​(3x⁰y^-5)³ =

​=

Multiple Choice

Simplify: (2w³)⁵

• 32w¹⁵

• 10w¹⁵

• 10w⁸

• 32w⁸

Multiple Choice

Simplify: (3x^-2y⁹)³

• 27x^-6y²⁷

• 9x^-6y²⁷

• 9x¹y¹²

• 27x¹y¹²