Question

Simplify. Assume p and q are greater than or equal to zero.
√20p^5q^10

Answer 1

To simplify the expression \(\sqrt{20p^5q^{10}}\), we can break it down into its components.

1. **Factor the expression inside the square root**:

\[
\sqrt{20p^5q^{10}} = \sqrt{20} \cdot \sqrt{p^5} \cdot \sqrt{q^{10}}
\]

2. **Simplify each component**:

- For \(\sqrt{20}\):
\[
20 = 4 \cdot 5 \quad \text{thus} \quad \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}
\]

- For \(\sqrt{p^5}\):
\[
p^5 = p^4 \cdot p = (p^2)^2 \cdot p \quad \text{thus} \quad \sqrt{p^5} = \sqrt{(p^2)^2 \cdot p} = p^2\sqrt{p}
\]

- For \(\sqrt{q^{10}}\):
\[
q^{10} = (q^5)^2 \quad \text{thus} \quad \sqrt{q^{10}} = q^5
\]

3. **Combine the results**:

Now putting it all together:
\[
\sqrt{20p^5q^{10}} = 2\sqrt{5} \cdot p^2\sqrt{p} \cdot q^5
\]

Rearranging gives us:
\[
= 2p^2q^5\sqrt{5p}
\]

Thus, the simplified form of \(\sqrt{20p^5q^{10}}\) is:
\[
\boxed{2p^2q^5\sqrt{5p}}
\]