Use Properties of Integers and Exponents

Properties of Integer Exponents

8.EE.1

Product of Powers

•  $a^m\cdot a^{n\ }=a^{m+n}$  

Power to a power

•  $\left(a^m\right)^n=a^{m\cdot n}$  

•  $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$  

•  $\left(ab\right)^n=a^nb^n$  

•  $\left(a^mb\right)^n=a^{m\cdot n}b^n$  

Quotient of powers

•  $\frac{a^m}{a^n}=a^{m-n}$  

Zero Exponent

• any number to the zero power equals 1

•  $a^0=1$  

•  $\left(-a\right)^0=1$  

•  $-\left(a\right)^0=-a^0=-1$  

Negative Exponents

• negative exponents tell you to take the reciprocal of the base

• negative exponents also mean to divide by the base

•  $a^{-n}=\frac{1}{a^n}$  

•  $\frac{1}{a^{-n}}=a^n$  

•  $\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n$  

Multiple Choice

Enter the value of n for the equation 

$$2^5\cdot2^n=2^{15}$$ . 

• 3

• 10

• -10

• $$\frac{1}{10}$$  

Multiple Select

Which of the following expressions is equivalent to

$$\frac{2^2}{2^8}$$ ?  Select all that apply.

• $$-2^4$$  

• $$2^{-6}$$  

• $$\left(\frac{1}{2}\right)^4$$  

• $$\frac{1}{64}$$  

• $$-\left(\frac{1}{2}\right)^6$$  

Multiple Select

Select ALL expressions equivalent to

$$\left(4^5\cdot4^{-3}\right)^{-2}$$  .

• $$\frac{1}{256}$$  

• 256

• $$4^{-10}\cdot4^6$$  

• $$4^{-3}\cdot4^{-5}$$  

Multiple Select

Which of these equations is correct? Select all that apply.

• $$\frac{2^4}{2^8}=\frac{1}{16}$$

• $$\frac{3^6}{3^4}=\frac{1}{9}$$

• $$\frac{4^9}{4^3}=64$$

• $$\frac{5^8}{5^6}=25$$

Multiple Select

Which of these equations is correct? Select all that apply.

• $$2^{-2}\cdot2^8=64$$

• $$3^{-5}\cdot3^1=\frac{1}{81}$$

• $$4^4\cdot4^{-3}=4$$

• $$5^3\cdot5^{-1}=\frac{1}{125}$$