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Operations on Rational and Irrational Numbers

Express the f...

Question

Express the following with rational denominator:

$(i i i) \frac{16}{\sqrt{41} - 5}$

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Solution

Given: $\frac{16}{\sqrt{41} - 5}$
Multiplying and dividing given number by
$\sqrt{41} + 5$
$= \frac{16 (\sqrt{41} + 5)}{(\sqrt{41} - 5) (\sqrt{41} + 5)}$
$= \frac{16 (\sqrt{41} + 5)}{(\sqrt{41})^{2} - (5)^{2}}$
Using identity $(a + b) (a - b) = a^{2} - b^{2}$
$= \frac{16 (\sqrt{41} + 5)}{41 - 25}$
$= \frac{16 (\sqrt{41} + 5)}{16}$
$= \sqrt{41} + 5$

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Similar questions

Express each one of the following with rational denominator :

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

(ii) $\frac{1}{\sqrt{6} - \sqrt{5}}$

(iii) $\frac{16}{\sqrt{41} - 5}$

(iv) $\frac{30}{5 \sqrt{3} - 3 \sqrt{5}}$

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

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

(vii) $\frac{6 - 4 \sqrt{2}}{6 + 4 \sqrt{2}}$

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

(ix) $\frac{b^{2}}{\sqrt{a^{2} + b^{2} + a}}$

\frac{16}{\sqrt{41} - 5}

(i i) \frac{1}{\sqrt{6} - \sqrt{5}}

Rationalize the denominator $\frac{16}{\sqrt{41} - 5}$ .

1) 6 : 8

2) 21 : 24

3) 33 : 77

4) 25 : 125

5) \frac{2}{3} : \frac{3}{5}

6) \frac{4}{5} : \frac{3}{8}

7) 1 \frac{1}{2} : 4 \frac{1}{2}

8) \frac{1}{3} : 2

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