Question

Simplest radical form of radical 405 over top radical 63

Answer 1

To simplify \(\frac{\sqrt{405}}{\sqrt{63}}\), we can first rewrite this as a single square root:

\[
\frac{\sqrt{405}}{\sqrt{63}} = \sqrt{\frac{405}{63}}
\]

Next, we simplify the fraction \(\frac{405}{63}\). We start by finding the greatest common divisor (GCD) of 405 and 63.

1. Factor 405:
- \(405 = 5 \times 81 = 5 \times 3^4\)
- Therefore, \(405 = 5 \times 3^4\).

2. Factor 63:
- \(63 = 7 \times 9 = 7 \times 3^2\)
- Therefore, \(63 = 7 \times 3^2\).

Now let's find the GCD. The common factor is \(3^2\).

Now, we can divide both the numerator and the denominator by their GCD:
\[
\frac{405 \div 9}{63 \div 9} = \frac{45}{7}
\]

Thus, we have:
\[
\sqrt{\frac{405}{63}} = \sqrt{\frac{45}{7}} = \frac{\sqrt{45}}{\sqrt{7}}
\]

Next, let's simplify \(\sqrt{45}\):
- \(45 = 9 \times 5 = 3^2 \times 5\)

Thus,
\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
\]

Putting it all together:
\[
\frac{\sqrt{45}}{\sqrt{7}} = \frac{3\sqrt{5}}{\sqrt{7}} = 3 \cdot \frac{\sqrt{5}}{\sqrt{7}} = 3 \sqrt{\frac{5}{7}}
\]

Thus, the simplest radical form of \(\frac{\sqrt{405}}{\sqrt{63}}\) is:

\[
3 \sqrt{\frac{5}{7}}
\]