Question

1. \\(\frac{sqrt{25}}{sqrt{15}}\\)\
2. \\(\frac{sqrt{5}}{sqrt{15}}\\)\
3. \\(\frac{sqrt{10}}{sqrt{6}}\\)\
4. \\(\frac{sqrt{15}}{sqrt{6}}\\)\
5. \\(\frac{sqrt{8}}{sqrt{6}}\\)\
6. \\(\frac{sqrt{6}}{sqrt{15}}\\)\
7. \\(\frac{3sqrt{3}}{sqrt{5}}\\)\
8. \\(\frac{3sqrt{3}}{5sqrt{2}}\\)\
9. \\(\frac{4}{3sqrt{5}}\\)\
10. \\(\frac{2sqrt{2}}{2sqrt{3}}\\)\
11. \\(\frac{5sqrt{4}}{sqrt{5}}\\)\
12. \\(\frac{3sqrt{5}}{2sqrt{2}}\\)

Answer:

Let's solve each problem step by step using the rule of rationalizing the denominator (or simplifying the square root fraction) which is $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ and for rationalizing, we multiply numerator and denominator by the denominator's square root to eliminate the square root from the denominator.

Problem 13: $\frac{\sqrt{25}}{\sqrt{15}}$

Step 1: Simplify $\sqrt{25}$

$\sqrt{25} = 5$, so the expression becomes $\frac{5}{\sqrt{15}}$

Step 2: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{15}$: $\frac{5\times\sqrt{15}}{\sqrt{15}\times\sqrt{15}}=\frac{5\sqrt{15}}{15}$

Step 3: Simplify the fraction

Divide numerator and denominator by 5: $\frac{\sqrt{15}}{3}$

Problem 14: $\frac{\sqrt{5}}{\sqrt{15}}$

Step 1: Use the square root fraction rule

$\frac{\sqrt{5}}{\sqrt{15}}=\sqrt{\frac{5}{15}}=\sqrt{\frac{1}{3}}$

Step 2: Rationalize the square root

$\sqrt{\frac{1}{3}}=\frac{\sqrt{1}}{\sqrt{3}}=\frac{1}{\sqrt{3}}$

Step 3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{3}$: $\frac{1\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{\sqrt{3}}{3}$

Problem 15: $\frac{\sqrt{10}}{\sqrt{6}}$

Step 1: Use the square root fraction rule

$\frac{\sqrt{10}}{\sqrt{6}}=\sqrt{\frac{10}{6}}=\sqrt{\frac{5}{3}}$

Step 2: Rationalize the square root

$\sqrt{\frac{5}{3}}=\frac{\sqrt{5}}{\sqrt{3}}$

Step 3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{3}$: $\frac{\sqrt{5}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{\sqrt{15}}{3}$

Problem 16: $\frac{\sqrt{15}}{\sqrt{6}}$

Step 1: Use the square root fraction rule

$\frac{\sqrt{15}}{\sqrt{6}}=\sqrt{\frac{15}{6}}=\sqrt{\frac{5}{2}}$

Step 2: Rationalize the square root

$\sqrt{\frac{5}{2}}=\frac{\sqrt{5}}{\sqrt{2}}$

Step 3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{2}$: $\frac{\sqrt{5}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{10}}{2}$

Problem 17: $\frac{\sqrt{8}}{\sqrt{6}}$

Step 1: Simplify $\sqrt{8}$

$\sqrt{8} = 2\sqrt{2}$, so the expression becomes $\frac{2\sqrt{2}}{\sqrt{6}}$

Step 2: Use the square root fraction rule

$\frac{2\sqrt{2}}{\sqrt{6}}=2\sqrt{\frac{2}{6}}=2\sqrt{\frac{1}{3}}$

Step 3: Rationalize the square root

$2\sqrt{\frac{1}{3}}=\frac{2}{\sqrt{3}}$

Step 4: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{3}$: $\frac{2\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{2\sqrt{3}}{3}$

Problem 18: $\frac{\sqrt{6}}{\sqrt{15}}$

Step 1: Use the square root fraction rule

$\frac{\sqrt{6}}{\sqrt{15}}=\sqrt{\frac{6}{15}}=\sqrt{\frac{2}{5}}$

Step 2: Rationalize the square root

$\sqrt{\frac{2}{5}}=\frac{\sqrt{2}}{\sqrt{5}}$

Step 3: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{5}$: $\frac{\sqrt{2}\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}=\frac{\sqrt{10}}{5}$

Problem 19: $\frac{3\sqrt{3}}{\sqrt{5}}$

Step 1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{5}$: $\frac{3\sqrt{3}\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}=\frac{3\sqrt{15}}{5}$

Problem 20: $\frac{3\sqrt{3}}{5\sqrt{2}}$

Step 1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{2}$: $\frac{3\sqrt{3}\times\sqrt{2}}{5\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{6}}{5\times2}=\frac{3\sqrt{6}}{10}$

Problem 21: $\frac{4}{3\sqrt{5}}$

Step 1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{5}$: $\frac{4\times\sqrt{5}}{3\sqrt{5}\times\sqrt{5}}=\frac{4\sqrt{5}}{3\times5}=\frac{4\sqrt{5}}{15}$

Problem 22: $\frac{2\sqrt{2}}{2\sqrt{3}}$

Step 1: Simplify the fraction by dividing numerator and denominator by 2

$\frac{\sqrt{2}}{\sqrt{3}}$

Step 2: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{3}$: $\frac{\sqrt{2}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{\sqrt{6}}{3}$

Problem 23: $\frac{5\sqrt{4}}{\sqrt{5}}$

Step 1: Simplify $\sqrt{4}$

$\sqrt{4} = 2$, so the expression becomes $\frac{5\times2}{\sqrt{5}}=\frac{10}{\sqrt{5}}$

Step 2: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{5}$: $\frac{10\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}=\frac{10\sqrt{5}}{5}=2\sqrt{5}$

Problem 24: $\frac{3\sqrt{5}}{2\sqrt{2}}$

Step 1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{2}$: $\frac{3\sqrt{5}\times\sqrt{2}}{2\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{10}}{2\times2}=\frac{3\sqrt{10}}{4}$

Final Answers:

1. $\boldsymbol{\frac{\sqrt{15}}{3}}$
2. $\boldsymbol{\frac{\sqrt{3}}{3}}$
3. $\boldsymbol{\frac{\sqrt{15}}{3}}$
4. $\boldsymbol{\frac{\sqrt{10}}{2}}$
5. $\boldsymbol{\frac{2\sqrt{3}}{3}}$
6. $\boldsymbol{\frac{\sqrt{10}}{5}}$
7. $\boldsymbol{\frac{3\sqrt{15}}{5}}$
8. $\boldsymbol{\frac{3\sqrt{6}}{10}}$
9. $\boldsymbol{\frac{4\sqrt{5}}{15}}$
10. $\boldsymbol{\frac{\sqrt{6}}{3}}$
11. $\boldsymbol{2\sqrt{5}}$
12. $\boldsymbol{\frac{3\sqrt{10}}{4}}$