Answered

a) [tex]\sqrt{\frac{81}{36}} =[/tex]
b) [tex]\sqrt{\frac{25}{9}} =[/tex]
c) [tex]\sqrt{\frac{125}{27}} =[/tex]
d) [tex]\sqrt[3]{\frac{8}{9}} =[/tex]
e) [tex]\sqrt[3]{\frac{169}{64}} =[/tex]
f) [tex]\sqrt[3]{\frac{64}{27}} =[/tex]

When the division results in an exact number:
[tex]\sqrt{6} : \sqrt{2} = \sqrt{\frac{6}{2}} = \sqrt{3}[/tex]
[tex]\sqrt{20} : \sqrt{10} = \sqrt{\frac{20}{10}} = \sqrt{2}[/tex]



Answer :

GinnyAnswer
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]