Sure, let's work through the given mathematical problem step by step.
1. Simplify the fraction:
We start with the fraction [tex]\(\frac{36}{81}\)[/tex].
- The fraction [tex]\(\frac{36}{81}\)[/tex] can be simplified by finding the greatest common divisor (GCD) of 36 and 81.
- The GCD of 36 and 81 is 9.
- Dividing both the numerator and the denominator by their GCD, we get:
[tex]\[
\frac{36 \div 9}{81 \div 9} = \frac{4}{9}
\][/tex]
2. Calculate the decimal form of the simplified fraction:
We can convert the fraction [tex]\(\frac{4}{9}\)[/tex] into decimal form:
- Performing the division [tex]\(4 \div 9\)[/tex] gives us:
[tex]\[
\frac{4}{9} = 0.4444444444444444 \, (\text{repeating})
\][/tex]
3. Find the square root of the fraction:
Next, we need to find the square root of the fraction [tex]\(\frac{4}{9}\)[/tex] or its decimal equivalent 0.4444444444444444.
- The square root of [tex]\(\frac{4}{9}\)[/tex] can be calculated as follows:
[tex]\[
\sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3}
\][/tex]
- Converting [tex]\(\frac{2}{3}\)[/tex] to decimal form, we get:
[tex]\[
\frac{2}{3} = 0.6666666666666666 \, (\text{repeating})
\][/tex]
Therefore, the simplified solution step by step is:
[tex]\[
\sqrt{\frac{36}{81}} = \sqrt{\frac{4}{9}} = \frac{2}{3} = 0.6666666666666666
\][/tex]
