Question

Multiply. Assume p is greater than or equal to zero, and write your answer in simplest form.
√6p^5 x √15p^3

Answer 1

To multiply the expressions \(\sqrt{6p^5}\) and \(\sqrt{15p^3}\), you can use the property of square roots that states:

\[
\sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b}
\]

Applying this property, we have:

\[
\sqrt{6p^5} \times \sqrt{15p^3} = \sqrt{(6p^5)(15p^3)}
\]

Now, calculate the product inside the square root:

\[
6 \times 15 = 90
\]

And for the \(p\) terms:

\[
p^5 \times p^3 = p^{5+3} = p^8
\]

So, we have:

\[
\sqrt{90p^8}
\]

Next, we can simplify \(\sqrt{90}\) and \(\sqrt{p^8}\):

1. For \(\sqrt{90}\):

- The prime factorization of 90 is \(90 = 9 \times 10 = 3^2 \times 10\),
- Therefore, \(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\).

2. For \(\sqrt{p^8}\):

- Since \(p^8\) is a perfect square, \(\sqrt{p^8} = p^{8/2} = p^4\).

Combining these results, we have:

\[
\sqrt{90p^8} = \sqrt{90} \cdot \sqrt{p^8} = 3\sqrt{10} \cdot p^4
\]

Thus, the final result in simplest form is:

\[
\boxed{3p^4\sqrt{10}}
\]