Question

Simplify √7 x √8, leaving your answer in exact form.

Answer 1

To simplify \( \sqrt{7} \times \sqrt{8} \), we can use the property of square roots that states:

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

Applying this to our expression:

\[
\sqrt{7} \times \sqrt{8} = \sqrt{7 \times 8}
\]

Calculating \( 7 \times 8 \):

\[
7 \times 8 = 56
\]

So we have:

\[
\sqrt{7} \times \sqrt{8} = \sqrt{56}
\]

Next, we can simplify \( \sqrt{56} \). First, we find the prime factorization of 56:

\[
56 = 7 \times 8 = 7 \times 4 \times 2 = 7 \times 2^3
\]

Now, we can write \( \sqrt{56} \) as:

\[
\sqrt{56} = \sqrt{7 \times 2^3} = \sqrt{7} \times \sqrt{2^3}
\]

The square root of \( 2^3 \) can be simplified:

\[
\sqrt{2^3} = \sqrt{2^2 \times 2} = \sqrt{2^2} \times \sqrt{2} = 2 \times \sqrt{2}
\]

Combining this back, we have:

\[
\sqrt{56} = \sqrt{7} \times (2 \times \sqrt{2}) = 2\sqrt{7} \times \sqrt{2}
\]

This can be rearranged as:

\[
2 \sqrt{14}
\]

Thus, the final answer, in exact form, is:

\[
\boxed{2\sqrt{14}}
\]