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Irrational Numbers

Express each ...

Question

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

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

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

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

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Solution

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

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

Hence the given expression is simplified with rational denominator to.

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(v) We know that rationalization factor for is.We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(viii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(ix) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to .

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

Q. Express each of the following as a rational number with positive denominator:
(i)

\frac{- 15}{- 28}

\frac{6}{- 9}

\frac{- 28}{- 11}

\frac{19}{- 7}

Express each of the following ratios as percents:
(i) $4 : 5$
(ii) $1 : 5$
(iii) $11 : 125$

Q. Write each of the following division as fraction:

(i) 6 ÷ 3
(ii) 25 ÷ 5
(iii) 125 ÷ 50
(iv) 55 ÷ 11

Find the multiplicative inverse (reciprocal) of each of the following rational numbers:

(i) 9 (ii)-7

(iii) $\frac{12}{5}$ (iv) $\frac{- 7}{9}$

(v) $\frac{- 3}{- 5}$ (vi) $\frac{2}{3} \times \frac{9}{4}$

(vii) $\frac{- 5}{8} \times \frac{16}{15}$ (viii) $- 2 \times \frac{- 3}{5}$

(ix) -1 (x) $\frac{0}{3}$

(ix)-1 (xi)1

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