Problem 90 - College Algebra

Question

Evaluate each expression without using a calculator. $$16^{-\frac{5}{2}}$$

Step-by-Step Solution

Answer

\(\frac{1}{1024}\)

Step 1: Deal with the Negative Exponent

According to the rules of exponents, any number (except 0) with a negative exponent can be written as the reciprocal with positive exponent. Meaning, \(16^{-\frac{5}{2}}\) can be written as \( \frac{1}{16^{\frac{5}{2}}} \)

Step 2: Break Down the Fractional Exponent

The exponent \(\frac{5}{2}\) can be dealt with in two steps: raise 16 to the 5th power, then take the square root. However, it is easier to first take the square root of 16 (since it is a perfect square) and then raise to the 5th power. This means the previous expression becomes: \( \frac{1}{(16^{\frac{1}{2}})^5} = \frac{1}{4^5} \)

Step 3: Calculate the expression

The number raised to the power value of 5 can be computed as multiplying the number by itself five times. It denotes \(4 \times 4 \times 4 \times 4 \times 4\) which will result in 1024. So the final simplified form of the expression is: \( \frac{1}{1024} \)