Question

\frac{\sqrt{x}}{128} = \frac{x}{\sqrt{512}}
Find x

Answer 1

To find \( x \) in the equation

\[
\frac{\sqrt{x}}{128} = \frac{x}{\sqrt{512}},
\]

we can start by simplifying the right side of the equation. We know that

\[
\sqrt{512} = \sqrt{256 \cdot 2} = \sqrt{256} \cdot \sqrt{2} = 16\sqrt{2}.
\]

Now substituting this back into the equation gives:

\[
\frac{\sqrt{x}}{128} = \frac{x}{16\sqrt{2}}.
\]

Next, we can cross-multiply to eliminate the fractions:

\[
\sqrt{x} \cdot 16 \sqrt{2} = x \cdot 128.
\]

This simplifies to:

\[
16 \sqrt{2} \sqrt{x} = 128 x.
\]

We can divide both sides of the equation by 16:

\[
\sqrt{2} \sqrt{x} = 8 x.
\]

Next, we can solve for \( \sqrt{x} \) by dividing both sides by \( \sqrt{x} \) (assuming \( \sqrt{x} \neq 0 \)):

\[
\sqrt{2} = 8 \frac{x}{\sqrt{x}} = 8 \sqrt{x}.
\]

Now we can solve for \( \sqrt{x} \):

\[
\sqrt{x} = \frac{\sqrt{2}}{8}.
\]

To find \( x \), we square both sides:

\[
x = \left(\frac{\sqrt{2}}{8}\right)^2 = \frac{2}{64} = \frac{1}{32}.
\]

Thus, the value of \( x \) is

\[
\boxed{\frac{1}{32}}.
\]