Question

c) $5\sqrt{5} = 125^{\frac{x}{4}}$
$5\sqrt{5} = (5^3)^{\frac{x}{4}}$
$5 \cdot 5^{\frac{1}{2}} = 5^{\frac{3x}{4}}$
$1 \cdot \frac{1}{2} = \frac{3x}{4}$
$4 \cdot 2 = 3x$
$8 = 3x$
$x = \frac{8}{3}$
answer: ______________
d) 2

Explanation:

Step1: Rewrite terms with base 5

Rewrite \(5\sqrt{5}\) as \(5\times5^{\frac{1}{2}} = 5^{1 + \frac{1}{2}}=5^{\frac{3}{2}}\) and \(125^{\frac{x}{4}}\) as \((5^{3})^{\frac{x}{4}} = 5^{\frac{3x}{4}}\).

Step2: Equate exponents

Since the bases are equal (\(5\)), set the exponents equal: \(\frac{3}{2}=\frac{3x}{4}\).

Step3: Solve for \(x\)

Multiply both sides by 4: \(3\times2 = 3x\) (simplifying \(\frac{3}{2}\times4 = 6\) and \(\frac{3x}{4}\times4 = 3x\), wait, earlier step had a miscalculation. Wait, original step: \(1+\frac{1}{2}=\frac{3}{2}\), then \(\frac{3}{2}=\frac{3x}{4}\). Multiply both sides by 4: \(3\times2=3x\) → \(6 = 3x\) → \(x = 2\). Wait, the handwritten solution had an error. Let's correct:

Correct Step1: \(5\sqrt{5}=5\times5^{\frac{1}{2}} = 5^{1+\frac{1}{2}}=5^{\frac{3}{2}}\), \(125^{\frac{x}{4}}=(5^{3})^{\frac{x}{4}}=5^{\frac{3x}{4}}\).

Correct Step2: Set exponents: \(\frac{3}{2}=\frac{3x}{4}\).

Correct Step3: Multiply both sides by 4: \(3\times2 = 3x\) → \(6 = 3x\) → \(x = 2\). (The handwritten step had a mistake in \(4\times2\), it should be \(\frac{3}{2}\times4 = 6\), not 8. So the correct \(x\) is 2.)

Answer:

\(x = 2\)