To express [tex]\(\sqrt{2 \frac{2}{9}}\)[/tex] in the form [tex]\(\frac{p}{q} \sqrt{r}\)[/tex], we will follow these steps:
1. Convert the mixed number to an improper fraction:
The mixed number [tex]\(2 \frac{2}{9}\)[/tex] can be converted to an improper fraction. First, multiply the whole number by the denominator and add the numerator:
[tex]\[
2 \frac{2}{9} = 2 + \frac{2}{9} = \frac{2 \times 9 + 2}{9} = \frac{18 + 2}{9} = \frac{20}{9}
\][/tex]
2. Calculate the square root of the improper fraction:
We need to find [tex]\(\sqrt{\frac{20}{9}}\)[/tex]. The square root of a fraction can be expressed as the square root of the numerator over the square root of the denominator:
[tex]\[
\sqrt{\frac{20}{9}} = \frac{\sqrt{20}}{\sqrt{9}}
\][/tex]
3. Simplify the square roots:
The square root of 9 is a perfect square and equals 3, so:
[tex]\[
\sqrt{\frac{20}{9}} = \frac{\sqrt{20}}{3}
\][/tex]
4. Express the square root of 20:
The square root of 20 can be simplified by factoring 20 into its prime factors:
[tex]\[
20 = 4 \times 5
\][/tex]
Since the square root of a product is the product of the square roots:
[tex]\[
\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}
\][/tex]
5. Combine the simplified terms:
Now, substituting [tex]\(\sqrt{20}\)[/tex] with [tex]\(2\sqrt{5}\)[/tex] in our expression:
[tex]\[
\sqrt{\frac{20}{9}} = \frac{2\sqrt{5}}{3}
\][/tex]
6. Write the final expression:
Now we have the form [tex]\(\frac{p}{q}\sqrt{r}\)[/tex] where [tex]\(p = 2\)[/tex], [tex]\(q = 3\)[/tex], and [tex]\(r = 5\)[/tex]:
[tex]\[
\sqrt{2 \frac{2}{9}} = \frac{2}{3} \sqrt{5}
\][/tex]
Thus, the expression [tex]\(\sqrt{2 \frac{2}{9}}\)[/tex] in the form [tex]\(\frac{p}{q} \sqrt{r}\)[/tex] is [tex]\(\frac{2}{3} \sqrt{5}\)[/tex].
