Question

Use the quotient rule to simplify the expression. Assume that x > 0.

squareroot 315x^6/squareroot 5x

squareroot 315x^6/squareroot 5x = blank

Answer 1

To simplify the expression \(\frac{\sqrt{315x^6}}{\sqrt{5x}}\), we can first use the property of square roots that states \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).

Applying this property, we have:

\[
\frac{\sqrt{315x^6}}{\sqrt{5x}} = \sqrt{\frac{315x^6}{5x}}
\]

Now, we can simplify the fraction inside the square root:

\[
\frac{315x^6}{5x} = \frac{315}{5} \cdot \frac{x^6}{x}
\]

Calculating \(\frac{315}{5}\):

\[
\frac{315}{5} = 63
\]

And simplifying \(\frac{x^6}{x}\):

\[
\frac{x^6}{x} = x^{6-1} = x^5
\]

Thus, we have:

\[
\sqrt{\frac{315x^6}{5x}} = \sqrt{63x^5}
\]

Next, we can separate this into its components:

\[
\sqrt{63x^5} = \sqrt{63} \cdot \sqrt{x^5}
\]

The square root of \(x^5\) can be simplified further:

\[
\sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x} = x^2 \cdot \sqrt{x}
\]

So we have:

\[
\sqrt{63x^5} = \sqrt{63} \cdot x^2 \cdot \sqrt{x}
\]

Finally, we can express \(\sqrt{63}\) in a simpler radical form. \(63 = 9 \cdot 7\), and we know that \(\sqrt{9} = 3\):

\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\]

Combining all the parts, we get:

\[
\sqrt{63x^5} = 3\sqrt{7} \cdot x^2 \cdot \sqrt{x} = 3x^2\sqrt{7x}
\]

Thus, the simplified expression is:

\[
\frac{\sqrt{315x^6}}{\sqrt{5x}} = 3x^2 \sqrt{7x}
\]

So the final answer is:

\[
\boxed{3x^2 \sqrt{7x}}
\]