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Simplify:i 5+...

Question

Simplify:

(i) $\frac{5 + \sqrt{3}}{5 - \sqrt{3}} + \frac{5 - \sqrt{3}}{5 + \sqrt{3}}$

(ii) $\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}}$

(iii) $\frac{2}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{3}{\sqrt{5} + \sqrt{2}}$

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Solution

(i) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

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Q. Simplify:

(i)

\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} + \frac{\sqrt{12}}{\sqrt{3} - \sqrt{2}}

\frac{5 + \sqrt{3}}{5 - \sqrt{3}} + \frac{5 - \sqrt{3}}{5 + \sqrt{3}}

\frac{7 + 3 \sqrt{5}}{3 + \sqrt{5}} + \frac{7 - 3 \sqrt{5}}{3 - \sqrt{5}}

\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}}

\frac{2}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{3}{\sqrt{5} + \sqrt{2}}

Verify the following

(i) $\frac{- 12}{5} + \frac{2}{7} = \frac{2}{7} + \frac{- 12}{5}$

(ii) $\frac{- 5}{8} + \frac{- 9}{13} = \frac{- 9}{13} + \frac{- 5}{8}$

(iii) $3 + \frac{- 7}{12} = \frac{- 7}{12} + 3$

(iv) $\frac{2}{- 7} + \frac{12}{- 35} = \frac{12}{- 35} + \frac{2}{- 7}$

Verify each of the following :

(i) $(\frac{5}{7} \times \frac{12}{13}) \times \frac{7}{18} = \frac{5}{7} \times (\frac{12}{13} \times \frac{7}{18})$
(ii) $\frac{- 13}{24} \times (\frac{- 12}{5} \times \frac{35}{36}) = (\frac{- 13}{24} \times \frac{- 12}{5}) \times \frac{35}{36}$
(iii) $(\frac{- 9}{5} \times \frac{- 10}{3}) \times \frac{21}{- 4} = \frac{- 9}{5} \times (\frac{- 10}{3} \times \frac{21}{- 4})$

Use the distributivity of multiplication of rational numbers over their addition to simplify:

(i) $\frac{3}{5} \times (\frac{35}{24} + \frac{10}{1})$

(ii) $\frac{- 5}{4} \times (\frac{8}{5} + \frac{16}{5})$

(iii) $\frac{2}{7} \times (\frac{7}{16} - \frac{21}{4})$

(iv) $\frac{3}{4} \times (\frac{8}{9} - 40)$

Q. Simplify:
(i)

\{{(\frac{1}{3})}^{- 3} - {(\frac{1}{2})}^{- 3}\} \div {(\frac{1}{4})}^{- 3}

(3^{2} - 2^{2}) \times {(\frac{2}{3})}^{- 3}

{\{{(\frac{1}{2})}^{- 1} \times (- 4)^{- 1}\}}^{- 1}

{[{\{{(\frac{- 1}{4})}^{2}\}}^{- 2}]}^{- 1}

{\{{(\frac{2}{3})}^{2}\}}^{3} \times {(\frac{1}{3})}^{- 4} \times 3^{- 1} \times 6^{- 1}

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