Question

What is the product of square root of, 5, end square root
5
​
and 10, square root of, 30, end square root10
30
​
in simplest radical form?

Answer 1

To find the product of \(\sqrt{5}\) and \(10\sqrt{30}\), we can multiply them together step by step.

1. Start by writing the expression:
\[
\sqrt{5} \times 10\sqrt{30}
\]

2. Rearrange the expression to group the constants and the radicals:
\[
10 \times \sqrt{5} \times \sqrt{30}
\]

3. Next, use the property of square roots that states \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\):
\[
= 10 \times \sqrt{5 \times 30}
\]

4. Now, calculate \(5 \times 30\):
\[
5 \times 30 = 150
\]

5. Substitute the value back into the radical:
\[
= 10 \times \sqrt{150}
\]

6. Simplify \(\sqrt{150}\):
- Factor \(150\): \(150 = 25 \times 6 = 5^2 \times 6\)
- Thus, \(\sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5\sqrt{6}\)

7. Substitute this back into the expression:
\[
10 \times \sqrt{150} = 10 \times 5\sqrt{6} = 50\sqrt{6}
\]

Thus, the product of \(\sqrt{5}\) and \(10\sqrt{30}\) in simplest radical form is:
\[
\boxed{50\sqrt{6}}
\]