Question

rationalize radicals
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simplify.

1. \\(\frac{\sqrt{10}}{\sqrt{125}}\\)
2. \\(\frac{\sqrt{3}}{\sqrt{16}}\\)
3. \\(\frac{\sqrt{20}}{\sqrt{9}}\\)
4. \\(\frac{\sqrt{5}}{\sqrt{80}}\\)
5. \\(\frac{2\sqrt{6}}{5\sqrt{8}}\\)
6. \\(\frac{5\sqrt{16}}{4\sqrt{9}}\\)
7. \\(\frac{2\sqrt{5}}{3\sqrt{45}}\\)
8. \\(\frac{2\sqrt{15}}{\sqrt{12}}\\)
9. \\(\frac{3\sqrt{8}}{4\sqrt{25}}\\)
10. \\(\frac{2\sqrt{25}}{3\sqrt{16}}\\)

Explanation:

Step1: Simplify \(\frac{\sqrt{10}}{\sqrt{125}}\)

Use the property \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\), so \(\frac{\sqrt{10}}{\sqrt{125}}=\sqrt{\frac{10}{125}}=\sqrt{\frac{2}{25}}=\frac{\sqrt{2}}{5}\)

Step2: Simplify \(\frac{\sqrt{3}}{\sqrt{16}}\)

\(\sqrt{16} = 4\), so \(\frac{\sqrt{3}}{\sqrt{16}}=\frac{\sqrt{3}}{4}\)

Step3: Simplify \(\frac{\sqrt{20}}{\sqrt{9}}\)

\(\sqrt{20}=2\sqrt{5}\), \(\sqrt{9} = 3\), so \(\frac{\sqrt{20}}{\sqrt{9}}=\frac{2\sqrt{5}}{3}\)

Step4: Simplify \(\frac{\sqrt{5}}{\sqrt{80}}\)

\(\sqrt{80}=4\sqrt{5}\), so \(\frac{\sqrt{5}}{\sqrt{80}}=\frac{\sqrt{5}}{4\sqrt{5}}=\frac{1}{4}\)

Step5: Simplify \(\frac{2\sqrt{6}}{5\sqrt{8}}\)

\(\sqrt{8}=2\sqrt{2}\), so \(\frac{2\sqrt{6}}{5\sqrt{8}}=\frac{2\sqrt{6}}{5\times2\sqrt{2}}=\frac{\sqrt{6}}{5\sqrt{2}}=\frac{\sqrt{12}}{10}=\frac{2\sqrt{3}}{10}=\frac{\sqrt{3}}{5}\)

Step6: Simplify \(\frac{5\sqrt{16}}{4\sqrt{9}}\)

\(\sqrt{16}=4\), \(\sqrt{9}=3\), so \(\frac{5\sqrt{16}}{4\sqrt{9}}=\frac{5\times4}{4\times3}=\frac{5}{3}\)

Step7: Simplify \(\frac{2\sqrt{5}}{3\sqrt{45}}\)

\(\sqrt{45}=3\sqrt{5}\), so \(\frac{2\sqrt{5}}{3\sqrt{45}}=\frac{2\sqrt{5}}{3\times3\sqrt{5}}=\frac{2}{9}\)

Step8: Simplify \(\frac{2\sqrt{15}}{\sqrt{12}}\)

\(\sqrt{12}=2\sqrt{3}\), so \(\frac{2\sqrt{15}}{\sqrt{12}}=\frac{2\sqrt{15}}{2\sqrt{3}}=\sqrt{5}\)

Step9: Simplify \(\frac{3\sqrt{8}}{4\sqrt{25}}\)

\(\sqrt{8}=2\sqrt{2}\), \(\sqrt{25}=5\), so \(\frac{3\sqrt{8}}{4\sqrt{25}}=\frac{3\times2\sqrt{2}}{4\times5}=\frac{6\sqrt{2}}{20}=\frac{3\sqrt{2}}{10}\)

Step10: Simplify \(\frac{2\sqrt{25}}{3\sqrt{16}}\)

\(\sqrt{25}=5\), \(\sqrt{16}=4\), so \(\frac{2\sqrt{25}}{3\sqrt{16}}=\frac{2\times5}{3\times4}=\frac{10}{12}=\frac{5}{6}\)

Answer:

s:

1. \(\boldsymbol{\frac{\sqrt{2}}{5}}\)
2. \(\boldsymbol{\frac{\sqrt{3}}{4}}\)
3. \(\boldsymbol{\frac{2\sqrt{5}}{3}}\)
4. \(\boldsymbol{\frac{1}{4}}\)
5. \(\boldsymbol{\frac{\sqrt{3}}{5}}\)
6. \(\boldsymbol{\frac{5}{3}}\)
7. \(\boldsymbol{\frac{2}{9}}\)
8. \(\boldsymbol{\sqrt{5}}\)
9. \(\boldsymbol{\frac{3\sqrt{2}}{10}}\)
10. \(\boldsymbol{\frac{5}{6}}\)