Question

evaluate without a calculator:
$16^{-3/4}$

Explanation:

Step1: Rewrite the base

Rewrite \( 16 \) as \( 2^4 \) or \( 4^2 \), here we use \( 16 = 2^4 \). So \( 16^{-3/4}=(2^4)^{-3/4} \).

Step2: Apply exponent rule

Using the rule \( (a^m)^n = a^{m\times n} \), we have \( (2^4)^{-3/4}=2^{4\times(-\frac{3}{4})} \).

Step3: Simplify the exponent

Calculate \( 4\times(-\frac{3}{4})=-3 \), so the expression becomes \( 2^{-3} \).

Step4: Apply negative exponent rule

Using \( a^{-n}=\frac{1}{a^n} \), \( 2^{-3}=\frac{1}{2^3} \).

Step5: Calculate the power

\( 2^3 = 8 \), so \( \frac{1}{2^3}=\frac{1}{8} \).

Answer:

\(\frac{1}{8}\)