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

Rather than simplifying the left side and matching exponents directly, we can test each candidate value of \( x \) by plugging it into \( 5^{x+2} \) and checking whether it equals the simplified left-hand side, which works out to a single power of 5.

1. x = 5: This gives \( 5^{x+2} = 5^{7} \). The left side, \( 5\sqrt{5} \times 5^3 \div 5^{-3/2} \), simplifies as \( 5^{1} \times 5^{1/2} \times 5^{3} \times 5^{3/2} = 5^{1+0.5+3+1.5} = 5^{6} \), not \( 5^{7} \), so this value does not satisfy the equation.
2. x = 7: This gives \( 5^{x+2} = 5^{9} \), far above the \( 5^{6} \) that the left side actually simplifies to, so this is too large.
3. x = 10: This gives \( 5^{x+2} = 5^{12} \), which is double the correct exponent of 6, clearly not a match.
4. x = 4: This gives \( 5^{x+2} = 5^{6} \), matching exactly what the left-hand side simplifies to.

Working out the left-hand side's exponent as \( 1 + \frac{1}{2} + 3 + \frac{3}{2} = 6 \) and comparing it against \( x+2 \) for each option shows that only \( x = 4 \) produces the required exponent of 6.

Therefore, the correct answer is 4.