Power of a Power Exponent Property

Power of Powers

Law of Exponents

Power of a Power

• When you have a number raised to a power that is then raised to another power it is called Power of a Power

• The inner exponent tells that I have two 4s being multiplied.

• The outer exponent tells me that these two fours being multiplied are also being multiplied three times.

• All together, how many 4s do we have?

Power of Power Shortcut?!?!?

When you have a power that is raised to another power, multiply the exponents!

(x²)⁵ = x^{2 * 5} = x¹⁰

Multiple Choice

(v⁶)³

• v^{6 + 3}

• v^{6 * 3}

• v^{6 - 3}

• v⁶

Multiple Choice

(3⁴)⁷

• 3²⁸

• 3³

• 3¹¹

• 12⁷

Multiple Choice

((-8)³)³

• (-8)⁶

• (-8)⁹

• (-8)⁰

• (24)³

Power of a Product

• If there is an exponent outside of the parentheses, it affects everything inside the parentheses.

• If we have two numbers being multiplied inside the parentheses and that group is raised to the power of three, then that group is repeated three times.

Power of Product Shortcut?!?!?

Instead of repeating it three times, you can just distribute the exponent to everything inside the group.

(xy)⁵ = x⁵y⁵

Further Example

• Now let's examine power of a product with power of a power.

• We know that if we have a power outside the parentheses we distribute it to everything inside the parentheses.

• We also know that if we distribute a power to another power we multiply those powers.

Multiple Choice

(1.2m)⁴

• 1.2m⁴

• 4.8m⁴

• 1.2⁴m⁴

• 1.2m⁵

Multiple Choice

(-3v²)⁴

• -3v⁶

• -3v⁸

• -12v⁶

• -3⁴v⁸

Multiple Choice

(2xy)⁵

• 2x⁵y⁶

• 32x⁵y⁵

• 2xy⁵

• 10x⁵y⁵

Multiple Choice

(3a²b⁴)³

• 3a⁵b⁷

• 6a⁶b¹²

• 27a⁶b¹²

• 9a⁵b⁷

Putting It All Together!

Can you simplify the following expressions using the different rules of exponents you have learned so far?

Multiple Choice

Simplify the expression: 2⁴ * 2⁵ - (2²)³

• 2⁹ - 2⁶

• 2²⁰ - 2⁵

• 2³

• 2²⁰- 2⁶

Multiple Choice

Simplify the expression:

5²(5³ * 5²)

• 5¹²

• 5⁶

• 5⁷

• 5¹⁰

Quotient of Powers

• We need to remember that when numbers are written in fraction form, we are actually dividing those two numbers.

• If two numbers with the same base but different exponents are being divided, we can simplify the expression.

• When you multiplied the same base you added the exponents.

• Since you are now dividing, you SUBTRACT the exponents!

Power of Quotient Shortcut?!?!

When dividing with the same base but different exponents, just SUBTRACT the exponents!

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

Multiple Choice

Simplify the expression: $\frac{6^{10}}{6^4}$  

•  $6^{14}$  

•  $6^6$  

•  $6^{40}$  

•  $6^2$  

Multiple Choice

Simplify the expression:  $\frac{8^9}{8^7}$  

•  $8^2$  

•  $8^3$  

•  $8^{16}$  

•  $8^{63}$  

Multiple Choice

Simplify the expression:  $\frac{\left(-3\right)^4}{\left(-3\right)^1}$  

•  $\left(-3\right)^2$  

•  $\left(-3\right)^4$  

•  $\left(-3\right)^5$  

•  $\left(-3\right)^3$  

Multiple Choice

Can you simplify this expression? 

 $\frac{3^4\cdot3^2}{3^3}$  

•  $3^{-1}$  

•  $3^3$  

•  $3^5$  

•  $3^2$  

Multiple Choice

One more challenge! Simplify this expression: $\frac{a^{10}}{a^6}\cdot\ \frac{a^7}{a^4}$  

•  $a^7$  

•  $a^{12}$  

•  $a^8$  

•  $a^{10}$