Certainly! Let's break down each part of the given problems step by step and solve them:
### Part a
[tex]\[
\sqrt{\frac{9}{16} \times \frac{4}{21}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{9}{16} \times \frac{4}{21} = \frac{9 \times 4}{16 \times 21} = \frac{36}{336}
\][/tex]
Now, simplify the fraction:
[tex]\[
\frac{36}{336} = \frac{1}{9}
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{\frac{1}{9}} = \frac{1}{3}
\][/tex]
### Part b
[tex]\[
\sqrt[3]{\frac{27}{8} \times \frac{1}{125}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{27}{8} \times \frac{1}{125} = \frac{27 \times 1}{8 \times 125} = \frac{27}{1000}
\][/tex]
Then, take the cube root:
[tex]\[
\sqrt[3]{\frac{27}{1000}} = \frac{\sqrt[3]{27}}{\sqrt[3]{1000}}
\][/tex]
Simplify the cube roots:
[tex]\[
\frac{\sqrt[3]{27}}{\sqrt[3]{1000}} = \frac{3}{10}
\][/tex]
### Part c
[tex]\[
\sqrt{\frac{1}{3} \times \frac{27}{4}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{1}{3} \times \frac{27}{4} = \frac{1 \times 27}{3 \times 4} = \frac{27}{12}
\][/tex]
Now, simplify the fraction:
[tex]\[
\frac{27}{12} = \frac{9}{4}
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}
\][/tex]
### Part d
[tex]\[
\sqrt{\frac{4}{16} \times \frac{9}{25}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{4}{16} \times \frac{9}{25} = \frac{4 \times 9}{16 \times 25} = \frac{36}{400}
\][/tex]
Now, simplify the fraction:
[tex]\[
\frac{36}{400} = \frac{9}{100}
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{\frac{9}{100}} = \frac{\sqrt{9}}{\sqrt{100}} = \frac{3}{10}
\][/tex]
### Part e
[tex]\[
\sqrt{\frac{6}{27} \times \frac{1}{6}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{6}{27} \times \frac{1}{6} = \frac{6 \times 1}{27 \times 6} = \frac{6}{162} = \frac{1}{27}
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{\frac{1}{27}} = \frac{1}{\sqrt{27}} = \frac{1}{3\sqrt{3}} = \frac{\sqrt{3}}{9}
\][/tex]
### Final expression
[tex]\[
\sqrt{\frac{2}{18} \times \frac{1}{49}}
\][/tex]
First, multiply the fractions:
[tex]\[
\frac{2}{18} \times \frac{1}{49} = \frac{2 \times 1}{18 \times 49} = \frac{2}{882}
\][/tex]
Now, simplify the fraction:
[tex]\[
\frac{2}{882} = \frac{1}{441}
\][/tex]
Then, take the square root:
[tex]\[
\sqrt{\frac{1}{441}} = \frac{1}{\sqrt{441}} = \frac{1}{21}
\][/tex]
So, the answers for each part are:
- Part a: [tex]\(\frac{1}{3}\)[/tex]
- Part b: [tex]\(\frac{3}{10}\)[/tex]
- Part c: [tex]\(\frac{3}{2}\)[/tex]
- Part d: [tex]\(\frac{3}{10}\)[/tex]
- Part e: [tex]\(\frac{\sqrt{3}}{9}\)[/tex]
- Final expression: [tex]\(\frac{1}{21}\)[/tex]
