SATHEE: Chapter 01 Number Systems Exercise-05

Chapter 01 Number Systems Exercise-05

EXERCISE 1.5

1. Find :

(i) $64^{\frac{1}{2}}$

(ii) $32^{\frac{1}{5}}$

(iii) $125^{\frac{1}{3}}$

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Solution

(i)

$\begin{aligned} 64^{\frac{1}{2}} & = (2^{6})^{\frac{1}{2}} & = 2^{2} = 4 \\ = 2^{6 \times \frac{1}{2}} & [(a^{m})^{n} = a^{m m!}]_{(16)^{\frac{3}{4}} = (2^{4})^{\frac{3}{4}}}^{(i i i)} \\ = 2^{3} = 8 \end{aligned}$

$= 2^{5 \times \frac{2}{3}} [(a^{m})^{n'} = a^{m m n}]$

2. Find:

(i) $9^{\frac{3}{2}}$

(ii) $32^{\frac{2}{5}}$

(iii) $16^{\frac{3}{4}}$

(iv) $125^{\frac{- 1}{3}}$

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Solution

(ii)

$\begin{aligned} 32^{\frac{1}{5}} & = (2^{5})^{\frac{1}{5}} \\ = (2)^{5 \times \frac{1}{5}} \\ = 2^{1} = 2 \end{aligned}$

$\begin{aligned} = 2^{4 \times \frac{3}{4}} \\ [(a^{m})^{n} = a^{m m}] \\ [(a^{m})^{n} = a^{m m}]_{(125)^{\frac{- 1}{3}}}^{(i v)} = \frac{1}{(125)^{\frac{1}{3}}} \\ = 2^{3} = 8 \end{aligned}$

$(125)^{\frac{1}{3}} = (5^{3})^{\frac{1}{3}}$

$\begin{matrix} = 5^{3 \times \frac{1}{3}} & [(a^{m})^{n} = a^{m m}] & = \frac{1}{5^{3 \times \frac{1}{3}}} \\ = 5^{1} = 5 & = \frac{1}{5} \end{matrix}$

$[a^{- m} = \frac{1}{a^{m}}]$

3. Simplify:

(i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$

(ii) ${(\frac{1}{3^{3}})}^{7}$

(iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$

(iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$

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Solution

(i)

$\begin{matrix} 2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}} & = 2^{\frac{2}{3} + \frac{1}{5}} & [a^{m} \cdot a^{n} = a^{m + n}] \\ = 2^{\frac{10 + 3}{15}} = 2^{\frac{13}{15}} \end{matrix}$

(ii)

$\begin{aligned} (\frac{1}{3^{3}})^{7} & = \frac{1}{3^{3 \times 7}} & [(a^{m})^{n} = a^{m m}] \\ = \frac{1}{3^{21}} & [\frac{1}{a^{m}} = a^{- m}] \end{aligned}$

(iii)

$\begin{matrix} \frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}} & = 11^{\frac{1}{2} - \frac{1}{4}} & [\frac{a^{m}}{a^{n}} = a^{m - n}] \\ = 11^{\frac{2 - 1}{4}} = 11^{\frac{1}{4}} \end{matrix}$

(iv)

$\begin{aligned} 7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}} & = (7 \times 8)^{\frac{1}{2}} [a^{m} \cdot b^{m} = (a b)^{m}] \\ = (56)^{\frac{1}{2}} \end{aligned}$

Chapter 01 Number Systems Exercise-05

EXERCISE 1.5

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Chapter 01 Number Systems Exercise-05

EXERCISE 1.5

Table of Contents

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Mentorship

NCERT Exercise Video Solution

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