Question

Use the product rule to simplify the following expression. Assume that variables represent nonnegative real
numbers.
square root x^13

square root x^13 blank (Please type an exact answer using radicals as needed.)

Answer 1

To simplify the expression \(\sqrt{x^{13}}\), we can use the property of square roots that states \(\sqrt{a^b} = a^{b/2}\).

Using this property, we have:

\[
\sqrt{x^{13}} = x^{13/2}
\]

Next, we can express \(x^{13/2}\) in terms of radicals. The exponent \(13/2\) can be split into an integer part and a fractional part:

\[
\frac{13}{2} = 6 + \frac{1}{2}
\]

Thus, we can express this as:

\[
x^{13/2} = x^6 \cdot x^{1/2} = x^6 \cdot \sqrt{x}
\]

Combining this, we can write the entire simplification as:

\[
\sqrt{x^{13}} = x^6 \sqrt{x}
\]

So the final answer is:

\[
x^6 \sqrt{x}
\]