Rational Exponents

Rational Exponents

Fractional Exponents. A rational exponent represents both an integer exponent and an nth root. 

Multiple Choice

Complete this sentence: To find the value of a number raised to the  $\frac{1}{2}$  power, find the ______ of the number.

Example:  $49^{\frac{1}{2}}=?$  

• Square Root

• 2nd Power

• nth root

Square Root

A square root of a number is a value that can be multiplied by itself to give the original number.

A square root of 49 is 7, because when 7 is multiplied by itself we get 49.

Multiple Choice

For any real number b,  $b^{\frac{1}{2}}=$  

•  $\sqrt{b}$  

•  $b^2$  

•  $2^b$  

Multiple Select

Pick all the right answers:
 $64^{\frac{1}{2}}=$  

•  $\sqrt{64}$  

•  $8$  

•  $8^2$  

•  $64$  

Multiple Select

Pick all the right answers:
 $\sqrt{25}=$  

•  $5$  

•  $\sqrt{5^2}$  

•  $25^{\frac{1}{2}}$  

•  $25$  

Multiple Choice

 $10^{\frac{1}{2}}=$  

•  $5$  

•  $\sqrt{10}$  

•  $\sqrt{5^2}$  

Multiple Choice

 $2g^{\frac{1}{2}}\ =\$  

•  $\sqrt{2g}$  

•  $2\sqrt{g}$  

•  $\sqrt{g}$  

•  $g$  

Multiple Select

Pick all the right answers:

 $\sqrt{16p}\ =\$  

•  $4p^{\frac{1}{2}}=$  

•  $\left(4p\right)^{\frac{1}{2}}$  

•  $\left(16p\right)^{\frac{1}{2}}$  

•  $4p$  

•  $4\sqrt{p}$  

nth root

This is read as the 5th root of 18

nth Roots

For any real numbers a and b and positive integer n:  $a^n\ =\ b$  then  $a\ =\ ^n\sqrt{b}$  

Example:  $5^3\ =\ 125\$  then  $5\ =\ ^3\sqrt{125}\ =\ \left(125\right)^{\frac{1}{3}}$ 

Simplify

1) $16^{\frac{3}{2}}=\left(^2\sqrt{16}\right)^3=\ ^2\sqrt{16^3}=$  

 $\ ^2\sqrt{4096}=4096^{\frac{1}{2}}=64$  


2) 
 $8^{\frac{5}{3}}=^3\sqrt{8^5}=^3\sqrt{32768}\ =$   $^3\sqrt{32^3}=\ 32$  

Multiple Choice

Write in radical form:  $17^{\frac{1}{3}}$  

•  $\ ^3\sqrt{17^1}$  

•  $\ ^1\sqrt{17^3}$  

•  $\ ^2\sqrt{17^3}$  

Multiple Choice

Write in radical form: 

 $m^{\frac{2}{3}}$  

•  $\ ^3\sqrt{m^2}$  

•  $\ ^2\sqrt{m^3}$  

Multiple Choice

Write in exponential form:

 $\ ^5\sqrt{h}$  

•  $h^{\frac{1}{5}}$  

•  $h^{\frac{5}{1}}$  

•  $h^{\frac{5}{2}}$  

•  $h^{\frac{2}{5}}$  

Multiple Choice

Write in exponential form:
 $\ ^3\sqrt{y^5}$  

•  $y^{\frac{5}{3}}$  

•  $y^{\frac{3}{5}}$  

Multiple Choice

Evaluate: $\ ^5\sqrt{243}$  

Hint: Get the prime factorization of 243.

•  $3$  

•  $5$  
  

•  $9\sqrt{3}$  

Multiple Choice

if $y^{\frac{1}{3}}=\ 10\$  then what is the value of y?

• 1,000

• 0.001

• 10

• 1