Question

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

Explanation:

Step1: Rewrite with same base

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 \(4\): \(3\times2 = 3x\) → \(6 = 3x\)? Wait, no, wait the original work had a miscalculation. Wait, \(5\sqrt{5}=5^{1}\times5^{\frac{1}{2}} = 5^{1+\frac{1}{2}}=5^{\frac{3}{2}}\), and \(125 = 5^{3}\), so \(125^{\frac{x}{4}}=(5^{3})^{\frac{x}{4}}=5^{\frac{3x}{4}}\). So equating exponents: \(\frac{3}{2}=\frac{3x}{4}\). Multiply both sides by \(4\): \(3\times2 = 3x\) → \(6 = 3x\) → \(x = 2\). Wait, the original work 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 \(\frac{3}{2}=\frac{3x}{4}\).

Correct Step3: Multiply both sides by \(4\): \(3\times2 = 3x\) → \(6 = 3x\) → \(x = 2\).

Wait, the original handwritten work had a mistake in the exponent of the left side. Let's redo properly:

Step1: Express both sides with base 5

Left side: \(5\sqrt{5}=5\times5^{\frac{1}{2}}=5^{1+\frac{1}{2}} = 5^{\frac{3}{2}}\) (since \(a^{m}\times a^{n}=a^{m + n}\)).

Right side: \(125^{\frac{x}{4}}=(5^{3})^{\frac{x}{4}}=5^{\frac{3x}{4}}\) (since \((a^{m})^{n}=a^{mn}\)).

Step2: Equate the exponents

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

Step3: Solve for \(x\)

Multiply both sides of the equation \(\frac{3}{2}=\frac{3x}{4}\) by \(4\) to get \(3\times2 = 3x\) → \(6 = 3x\). Then divide both sides by \(3\): \(x = 2\).

Wait, the original handwritten work had a mistake in the left - hand exponent calculation. The correct calculation is as above.

Answer:

The correct value of \(x\) is \(2\) (after correcting the error in the original work). If we follow the original (incorrect) steps, but the correct mathematical process gives \(x = 2\).