Question

Sqrt(128/x)=x/sqrt(512)…find x

A)14
B)16
C)24
D)18

Answer 1

To solve the equation \(\sqrt{\frac{128}{x}} = \frac{x}{\sqrt{512}}\), we can first simplify the equation.

**Step 1: Rewrite the right side.**

We know that:
\[
\sqrt{512} = \sqrt{256 \times 2} = \sqrt{256} \cdot \sqrt{2} = 16\sqrt{2}
\]
Thus, we rewrite the equation as:
\[
\sqrt{\frac{128}{x}} = \frac{x}{16\sqrt{2}}
\]

**Step 2: Square both sides to eliminate the square root.**
\[
\frac{128}{x} = \frac{x^2}{512}
\]

**Step 3: Cross-multiply.**
\[
128 \cdot 512 = x^3
\]

**Step 4: Calculate \(128 \cdot 512\).**
\[
128 \cdot 512 = 128 \cdot (500 + 12) = 128 \cdot 500 + 128 \cdot 12
\]
\[
128 \cdot 500 = 64000
\]
\[
128 \cdot 12 = 1536
\]
Adding these together gives:
\[
64000 + 1536 = 65536
\]

Thus, we have:
\[
x^3 = 65536
\]

**Step 5: Find \(x\) by taking the cube root.**
\[
x = \sqrt[3]{65536}
\]

Since \(65536 = 16^4\), we can further simplify:
\[
x = 16
\]

Hence, \(x = 16\).

The option that matches is \(B) 16\).