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: Understanding the Question:
We need to simplify the expression: \[ 5\sqrt{5} \times 5^3 \div 5^{-3/2} \] and compare it with: \[ 5^{x+2} \] to find the value of \(x\).

Step 2: Key Formula or Approach:
Use the laws of exponents: \[ a^m \times a^n=a^{m+n} \] \[ \frac{a^m}{a^n}=a^{m-n} \] Also, \[ \sqrt{5}=5^{1/2} \]

Step 3: Detailed Explanation:
Rewrite the expression: \[ 5\sqrt{5} \times 5^3 \div 5^{-3/2} \] Since: \[ 5\sqrt{5}=5^1 \times 5^{1/2}=5^{3/2} \] So: \[ 5^{3/2}\times 5^3 \div 5^{-3/2} \] Apply exponent rules: \[ =5^{3/2+3-(-3/2)} \] \[ =5^{3/2+3+3/2} \] \[ =5^{3+3} \] \[ =5^6 \] Given: \[ 5^{x+2}=5^6 \] Equate exponents: \[ x+2=6 \] \[ x=4 \]

Step 4: Final Answer:
The value of \(x\) is: \[ \boxed{4} \] Hence, the correct option is: \[ \boxed{\text{(D) 4}} \]

Answer 2

Instead of manipulating the exponents symbolically from the start, we can evaluate the left-hand side numerically and then match it to a power of 5.

First, compute an approximate numeric value for each piece: \(5\sqrt{5} \approx 5 \times 2.236 = 11.18\), \(5^3=125\), and \(5^{-3/2} = \frac{1}{5^{3/2}} \approx \frac{1}{11.18} \approx 0.0894\).

Now combine them as the original expression states:

\[ 5\sqrt{5} \times 5^3 \div 5^{-3/2} \approx 11.18 \times 125 \div 0.0894 \approx 1397.5 \div 0.0894 \approx 15633 \]

This value is very close to \(5^6=15625\), with the small gap explained by rounding in the approximations above. So the left-hand side equals \(5^6\).

Since the equation states that this equals \(5^{x+2}\), and both sides use the same base 5, the exponents must match:

\[ x+2=6 \implies x=4 \]

Checking the other options: \(x=5\) would require the left side to equal \(5^7\), \(x=7\) would require \(5^9\), and \(x=10\) would require \(5^{12}\), none of which are anywhere close to the value computed numerically above.

Therefore, the correct answer is 4.