Question

How can you rewrite 9 1/3
using a root?

Answer 1

To rewrite the mixed number \( 9 \frac{1}{3} \) using a root, we first convert it to an improper fraction.

1. Convert \( 9 \frac{1}{3} \) to an improper fraction:
\[
9 \frac{1}{3} = 9 + \frac{1}{3} = \frac{9 \times 3 + 1}{3} = \frac{27 + 1}{3} = \frac{28}{3}
\]

2. Now, to express \( \frac{28}{3} \) using a root, we can express \( 28 \) as a product of its prime factors:
\[
28 = 4 \times 7 = 2^2 \times 7
\]
Thus, we can write \( \frac{28}{3} \) as:
\[
\frac{28}{3} = \frac{4 \times 7}{3} = \frac{2^2 \times 7}{3}
\]

3. This can be rewritten using a root by expressing \( 4 \) as \( 2^2 \):
\[
\frac{2^2 \times 7}{3} = \frac{\sqrt{4} \times 7}{3}
\]

Hence, \( 9 \frac{1}{3} \) can be rewritten as:
\[
\frac{\sqrt{4} \times 7}{3}
\]

Alternatively, we can express it as a single root:
\[
9 \frac{1}{3} = 28 \div 3 = \frac{\sqrt{28^2}}{3}
\]
This second expression uses the square root of \( 28 \) effectively.

So two possible forms to express \( 9 \frac{1}{3} \) using roots are:
\[
\frac{\sqrt{4} \times 7}{3} \quad \text{or} \quad \frac{\sqrt{28^2}}{3}
\]