Question

What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary.
8√27/2√45

Answer 1

To simplify the expression \(\frac{8\sqrt{27}}{2\sqrt{45}}\), we'll start by simplifying both the numerator and the denominator.

1. Simplify \(\sqrt{27}\):
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]

2. Simplify \(\sqrt{45}\):
\[
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
\]

Now, substituting back into the expression, we have:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8 \cdot 3\sqrt{3}}{2 \cdot 3\sqrt{5}}
\]

This simplifies further:
\[
= \frac{24\sqrt{3}}{6\sqrt{5}}
\]

Next, we simplify the fraction by dividing both the numerator and denominator by 6:
\[
= \frac{4\sqrt{3}}{\sqrt{5}}
\]

To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{5}\):
\[
= \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5}
\]

Thus, the expression \(\frac{8\sqrt{27}}{2\sqrt{45}}\) simplifies to:
\[
\boxed{\frac{4\sqrt{15}}{5}}
\]