Alright, let's go through each part step-by-step and find the required values for each.
### a) [tex]\(\sqrt{\frac{81}{36}}\)[/tex]
To find the square root of [tex]\(\frac{81}{36}\)[/tex], we first simplify the fraction:
[tex]\[ \frac{81}{36} = \frac{9}{4} \][/tex]
Now, take the square root of [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[ \sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2} = 1.5 \][/tex]
So,
[tex]\[ \sqrt{\frac{81}{36}} = 1.5 \][/tex]
### b) [tex]\(\sqrt{\frac{25}{9}}\)[/tex]
For this part, we simplify the fraction:
[tex]\[ \frac{25}{9} \][/tex]
Now, take the square root of [tex]\(\frac{25}{9}\)[/tex]:
[tex]\[ \sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3} \approx 1.6666666666666667 \][/tex]
So,
[tex]\[ \sqrt{\frac{25}{9}} = 1.6666666666666667 \][/tex]
### c) [tex]\(\sqrt{\frac{125}{27}}\)[/tex]
In this part, we simplify the fraction:
[tex]\[ \frac{125}{27} \][/tex]
Next, take the square root of [tex]\(\frac{125}{27}\)[/tex]:
[tex]\[ \sqrt{\frac{125}{27}} \approx 2.151657414559676 \][/tex]
So,
[tex]\[ \sqrt{\frac{125}{27}} \approx 2.151657414559676 \][/tex]
### d) [tex]\(\sqrt[3]{\frac{8}{9}}\)[/tex]
To find the cube root of [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[ \frac{8}{9} \][/tex]
Now, take the cube root of [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[ \sqrt[3]{\frac{8}{9}} \approx 0.9614997135382722 \][/tex]
So,
[tex]\[ \sqrt[3]{\frac{8}{9}} \approx 0.9614997135382722 \][/tex]
### e) [tex]\(\sqrt[3]{\frac{169}{64}}\)[/tex]
For this part, we simplify the fraction:
[tex]\[ \frac{169}{64} \][/tex]
Next, take the cube root of [tex]\(\frac{169}{64}\)[/tex]:
[tex]\[ \sqrt[3]{\frac{169}{64}} \approx 1.382193703419718 \][/tex]
So,
[tex]\[ \sqrt[3]{\frac{169}{64}} \approx 1.382193703419718 \][/tex]
### f) [tex]\(\sqrt[3]{\frac{64}{27}}\)[/tex]
Here, we simplify the fraction:
[tex]\[ \frac{64}{27} \][/tex]
Now, take the cube root of [tex]\(\frac{64}{27}\)[/tex]:
[tex]\[ \sqrt[3]{\frac{64}{27}} \approx 1.3333333333333333 \][/tex]
So,
[tex]\[ \sqrt[3]{\frac{64}{27}} \approx 1.3333333333333333 \][/tex]
Here are the final results:
[tex]\[
a) \sqrt{\frac{81}{36}}= 1.5
\][/tex]
[tex]\[
b) \sqrt{\frac{25}{9}}= 1.6666666666666667
\][/tex]
[tex]\[
c) \sqrt{\frac{125}{27}} \approx 2.151657414559676
\][/tex]
[tex]\[
d) \sqrt[3]{\frac{8}{9}} \approx 0.9614997135382722
\][/tex]
[tex]\[
e) \sqrt[3]{\frac{169}{64}} \approx 1.382193703419718
\][/tex]
[tex]\[
f) \sqrt[3]{\frac{64}{27}} \approx 1.3333333333333333
\][/tex]
