Question

The value of \( 8^{2/3} - 16^{1/4} - 9^{1/2} \) is:

• A. \( -1 \)
• B. \( -2 \)
• C. \( -3 \)
• D. \( -4 \)
• E. \( -5 \)

Hint:

Always convert numbers into prime powers first — it makes fractional exponents much easier to evaluate.

Answer 1

The Correct Option is A

Concept: Use the exponent rule: \[ (a^m)^n = a^{mn} \] It is often easier to rewrite numbers as powers of prime numbers before applying fractional exponents.

Step 1: Simplify each term separately.
1) Rewrite \( 8 \) as \( 2^3 \): \[ 8^{2/3} = (2^3)^{2/3} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4 \] 2) Rewrite \( 16 \) as \( 2^4 \): \[ 16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \] 3) Rewrite \( 9 \) as \( 3^2 \): \[ 9^{1/2} = (3^2)^{1/2} = 3^{2 \cdot \frac{1}{2}} = 3^1 = 3 \]

Step 2: Substitute the values back into the expression.
\[ 4 - 2 - 3 \]

Step 3: Perform final calculation.
\[ 4 - 2 = 2 \] \[ 2 - 3 = -1 \] \[ \boxed{-1} \]