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: Take \(\log_5\) of both sides of the equation \(5\sqrt{5}\times5^3\div5^{-3/2}=5^{x+2}\), which turns every power of 5 into just its exponent.

Step 2: On the left, \(\log_5(5\sqrt{5}) = \log_5(5)+\log_5(\sqrt{5}) = 1+\frac{1}{2} = \frac{3}{2}\), \(\log_5(5^3)=3\), and \(\log_5(5^{-3/2})=-\frac{3}{2}\). Since the original expression multiplies the first two and divides by the third, the logs add and subtract the same way: \(\frac{3}{2}+3-\left(-\frac{3}{2}\right)=6\).

Step 3: On the right, \(\log_5(5^{x+2})=x+2\). Setting the two sides equal: \(x+2=6\), so \(x=4\).
\[ \boxed{x=4} \]