Question

Find the cube root of 512:3sqrt512

Answer 1

To simplify the expression \(512:3\sqrt{512}\), we first need to calculate the cube root of \(512\).

1. **Find the cube root of \(512\)**:
\[
512 = 8^3
\]
Therefore, the cube root of \(512\) is:
\[
\sqrt[3]{512} = 8
\]

2. **Compute \(3\sqrt{512}\)**:
To find \(3\sqrt{512}\), we first need to simplify \(\sqrt{512}\):
\[
512 = 256 \times 2 = 16^2 \times 2
\]
Thus,
\[
\sqrt{512} = \sqrt{16^2 \times 2} = 16\sqrt{2}
\]

Now calculate \(3\sqrt{512}\):
\[
3\sqrt{512} = 3 \times 16\sqrt{2} = 48\sqrt{2}
\]

3. **Combine results**:
The expression \(512:3\sqrt{512}\) can be interpreted as:

\[
\frac{512}{3\sqrt{512}} = \frac{512}{48\sqrt{2}}
\]

To further simplify:
\[
\frac{512}{48\sqrt{2}} = \frac{512 \div 16}{48 \div 16 \sqrt{2}} = \frac{32}{3\sqrt{2}}
\]

4. **Rationalize the denominator**:
To rationalize:
\[
\frac{32}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{32\sqrt{2}}{3 \times 2} = \frac{32\sqrt{2}}{6} = \frac{16\sqrt{2}}{3}
\]

The result simplifies to:

\[
\boxed{\frac{16\sqrt{2}}{3}}
\]