Question

Rationalize the denominator and simplify.
StartFraction StartRoot 72 EndRoot Over StartRoot 7 EndRoot EndFraction
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Part 1
StartFraction StartRoot 72 EndRoot Over StartRoot 7 EndRoot EndFraction
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​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Answer 1

To rationalize the denominator of the expression \( \frac{\sqrt{72}}{\sqrt{7}} \), we can multiply both the numerator and the denominator by \( \sqrt{7} \) to eliminate the radical in the denominator.

Here are the steps:

1. **Multiply by \( \sqrt{7} \)**:
\[
\frac{\sqrt{72}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{72} \cdot \sqrt{7}}{7}
\]

2. **Combine the radicals in the numerator**:
\[
= \frac{\sqrt{72 \cdot 7}}{7}
\]

3. **Calculate \( 72 \cdot 7 \)**:
\[
72 \cdot 7 = 504
\]
So,
\[
= \frac{\sqrt{504}}{7}
\]

4. **Simplify \( \sqrt{504} \)**: First, let's factor \( 504 \):
\[
504 = 72 \cdot 7 = 36 \cdot 14 = 36 \cdot (2 \cdot 7) = (6^2) \cdot (2 \cdot 7) = 36 \cdot 14
\]

However, we can take \( 504 \) apart a bit further:
\[
504 = 2^3 \cdot 3^2 \cdot 7
\]
Taking the square root:
\[
\sqrt{504} = \sqrt{(2^2 \cdot 3^2) \cdot (2 \cdot 7)} = \sqrt{36 \cdot 14} = 6\sqrt{14}
\]

5. **Substituting back**:
\[
= \frac{6\sqrt{14}}{7}
\]

Thus, the simplified form of the expression is:
\[
\frac{6\sqrt{14}}{7}
\]

Final answer:
\[
\frac{6\sqrt{14}}{7}
\]