Question

If \[ 5\sqrt{5} \times 5^3 \div 5^{-3/2} = 5^{x+2}, \] find the value of \(x\).

• A. 5
• B. 7
• C. 10
• D. 4

Hint:

Convert roots into fractional exponents first. Then apply exponent laws carefully while multiplying or dividing powers with the same base.

Answer 1

The Correct Option is A

Step 1: Rewrite every term on the left as a power of 5: \( 5\sqrt{5} = 5^{3/2} \), and \( 5^{-3/2} \) stays as is, so the expression becomes \( 5^{3/2} \times 5^{3} \div 5^{-3/2} \).

Step 2: Taking \( \log_5 \) of both sides turns the multiplication and division into addition and subtraction of exponents: \( \frac{3}{2} + 3 - \left(-\frac{3}{2}\right) = \frac{3}{2} + 3 + \frac{3}{2} = 6 \).

Step 3: Since \( \log_5\left(5^{x+2}\right) = x+2 \), setting the two exponents equal gives \( x + 2 = 6 \), so \( x = 4 \).
\[ \boxed{x = 4} \]