Which sequence of steps correctly gives the value of `4^((3)/(2))` and algebraically justifies the values?

Options

A. `4^((3)/(2))=(4^(2))^((1)/(3))=sqrt(3)(4^(2))=sqrt(3)(16)`
B. `4^((3)/(2))=(4^(2))÷3=16÷3=(16)/(3)`
C. `4^((3)/(2))=(4^(2))÷2=64÷2=32`
D. `4^((3)/(2))=(4^(3))^((1)/(2))=sqrt(4^(3))=sqrt(64)=8`

Answer

C

Explanation

Difficult: Medium
Category: Passport to Advanced Math/ Exponents
Strategic Advice: Use rules of exponents to rewrite the expression. The saying "power over root" may help as well.
Getting to the Answer: Use the exponent rule `a^(mn)=(a^(m))^(n)` and the fact that `(3)/(2)=3times(1)/(2)` to rewrite the expression. Then simplify using the property `a^((1)/(2))=sqrt(a)`:
`4^((3)/(2))=(4^(3))^((1)/(2))=sqrt(4^(3))=sqrt(4times4times4)=sqrt(64)=8`
This means (D) is correct.