Ncert Solutions Class 8 Maths Chapter 12 In Hindi

प्रश्नावली 12.1

1. मान ज्ञात कीजिए:

(i) $3^{-2}$

उत्तर $3^{-2}=\left(\dfrac{1}{3^{2}}\right)=\left(\dfrac{1}{9}\right)$

(ii) $(-4)^{-2}$

उत्तर: $\quad(-4)^{-2}=\left(\dfrac{1}{-4^{2}}\right)=\left(\dfrac{1}{16}\right)$

(iii) $\left(\dfrac{1}{2}\right)^{-5}$

उत्तर: 

$\quad\left(\dfrac{1}{2}\right)^{-5}=\left\{\dfrac{1^{-5}}{2^{-5}}\right\}=\left\{\begin{array}{l}\left.{\dfrac{1}{\dfrac{1}{5}}}{\dfrac{1}{2^{5}}}\right\}=\left\{\dfrac{\dfrac{1}{1}}{\dfrac{1}{32}}\right\}=\dfrac{32}{1}=32\end{array}\right.$

2. सरल कीजिए एवं उत्तर को धनात्मक घातांक के रूप में व्यक्त कीजिए :

(i)$\quad(-4)^{5} \div(-4)^{8}$

Ans: $\quad(-4)^{5} \div(-4)^{8}=(-4)^{5-8}=(-4)^{-3}=\left(\dfrac{1}{-4^{3}}\right)$

(ii) $\left(\dfrac{1}{2^{3}}\right)^{2}$

Ans: $\left(\dfrac{1}{2^{3}}\right)^{2}=\left(2^{-3}\right)^{2}=2^{-3 \times 2}=2^{-6}$

(iii) $\quad(-3)^{4} \times\left(\dfrac{5}{3}\right)^{4}$

Ans: $\quad(-3)^{4} \times\left(\dfrac{5}{3}\right)^{4}=\left[-3 \times\left(\dfrac{5}{3}\right)\right]^{4}=[-5]^{4}$

(iv) $\left(3^{-7} \div 3^{-10}\right) \times 3^{-5}$

Ans: $\quad\left(3^{-7} \div 3^{-10}\right) \times 3^{-5}=\left[3^{-7+10} \times 3^{-5}\right]=\left[3^{3} \times 3^{-5}\right]=$ $3^{[3+(-5)]}=3^{-2}$

(v) $\quad 2^{-3} \times(-7)^{-3}$

Ans: $\quad 2^{-3} \times(-7)^{-3}=[2 \times(-7)]^{-3}=[-14]^{-3}=\left[\dfrac{1}{(-14)^{3}}\right]$

3. मान ज्ञात कीजिए :

(i) $\left(3^{0}+4^{-1}\right) \times 2^{2}$

उत्तर: (i) $\left(3^{0}+4^{-1}\right) \times 2^{2}=\left(1+\dfrac{1}{4^{1}}\right) \times 4=\left(1+\dfrac{1}{4}\right) \times 4=\left(\dfrac{5}{4}\right) \times 4=5$

(ii) $\quad\left(2^{-1} \times 4^{-1}\right) \div 2^{-2}$

उत्तर:(ii)$\left(2^{-1} \times 4^{-1}\right) \div 2^{-2}=(2 \times 4)^{-1} \div 4^{-1}$ $\left\{\left(\dfrac{1}{8^{1}}\right) \div\left(\dfrac{1}{2^{2}}\right)\right\}=\left\{\left(\dfrac{1}{8}\right) \div\left(\dfrac{1}{4}\right)\right\}=\dfrac{4}{8}=\dfrac{1}{2}$

(iii) $\left(\dfrac{1}{2}\right)^{-2}+\left(\dfrac{1}{3}\right)^{-2}+\left(\dfrac{1}{4}\right)^{-2}$

उत्तर:                                                                                                                                         (iii)$\left(\dfrac{1}{2}\right)^{-2}+\left(\dfrac{1}{3}\right)^{-2}+\left(\dfrac{1}{4}\right)^{-2}=\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\right)^{-2}=\left(\dfrac{6}{12}+\dfrac{4}{12}+\right.$ $\left.\dfrac{3}{12}\right)^{-2}=\left(\dfrac{13}{12}\right)^{-2}=\left(\dfrac{13^{-2}}{12^{-2}}\right)=\dfrac{\dfrac{1}{13^{2}}}{\dfrac{1}{12^{2}}}=\left\{\dfrac{1}{\dfrac{169}{144}}\right\}=\dfrac{144}{169}$ 

(iv) $\quad\left(3^{-1}+4^{-1}+5^{-1}\right)^{0}$

उत्तर: (iv) $\quad\left(3^{-1}+4^{-1}+5^{-1}\right)^{0}=1 \quad\left[a^{0}=1\right]$

(v) $\left\{\left(-\dfrac{2}{3}\right)^{-2}\right\}^{2}$

उत्तर: (v)$\left\{\left(-\dfrac{2}{3}\right)^{-2}\right\}^{2}=\left(-\dfrac{2}{3}\right)^{-2 \times 2}=\left(-\dfrac{2}{3}\right)^{-4}=\left(-\dfrac{2^{-4}}{3^{-4}}\right)=\left\{-\dfrac{\dfrac{1}{2^{4}}}{\dfrac{1}{3^{4}}}\right\}=$ $\left(-\dfrac{\dfrac{1}{16}}{\dfrac{1}{81}}\right)=\left(-\dfrac{81}{16}\right)$

4. मान ज्ञात कीजिए:

(i) $\dfrac{8^{-1} \times 5^{3}}{2^{-4}}$

उत्तर: (i) $\quad \dfrac{8^{-1} \times 5^{3}}{2^{-4}}=\dfrac{\left\{\left(\dfrac{1}{8^{1}}\right) \times 125\right\}}{\dfrac{1}{2^{4}}}=\dfrac{\dfrac{1}{8} \times 125}{\dfrac{1}{16}}=\dfrac{16 \times 125}{8}=2 \times 125=250$

(ii) $\left(5^{-1} \times 2^{-1}\right) \times 6^{-1}$

उत्तर:(ii) $\left(5^{-1} \times 2^{-1}\right) \times 6^{-1}-\left(5^{-1} \times 2^{-1} \times 6^{-1}\right)-(5 \times 2 \times 6)^{-1}$ $60^{-1}=\dfrac{1}{60}$

5. $m$ का मान ज्ञात कीजिए जिसके लिए $5^{m} \div 5^{-3}=5^{5}$

उत्तर: $5^{m} \div 5^{-3}=5^{5}$

तथा, $5^{m+3}=5^{5}$

,तथा $m+3=5$

तथा, $m=5-3=2$

तब, $m=2$ [proved]

6. मान ज्ञात कीजिए:

(i) $\left\{\left(\dfrac{1}{3}\right)^{-1}-\left(\dfrac{1}{4}\right)^{-1}\right\}^{-4}$

उत्तर:(i)$\left\{\left(\dfrac{1}{3}\right)^{-1}-\left(\dfrac{1}{4}\right)^{-1}\right\}^{-4}=\left\{\left(\dfrac{1}{3}-\dfrac{1}{4}\right)^{-1}\right\}^{-4}=\left\{\left(\dfrac{4-3}{12}\right)^{-1}\right\}^{-4}=$

$\left(\dfrac{1}{12}\right)^{\{(-1) \times(-4)\}}=\left(\dfrac{1}{12}\right)^{4}=\left(\dfrac{1^{4}}{12^{4}}\right)=\dfrac{1}{20746}$

(ii) $\left(\dfrac{5}{8}\right)^{-7} \times\left(\dfrac{8}{5}\right)^{-4}$

उत्तर: (ii) $\left(\dfrac{5}{8}\right)^{-7} \times\left(\dfrac{8}{5}\right)^{-4}=\dfrac{\dfrac{1}{5^{7}}}{\dfrac{1}{8^{7}}} \times \dfrac{\dfrac{1}{8^{4}}}{\dfrac{1}{5^{4}}}=\dfrac{8^{7}}{5^{7}} \times \dfrac{5^{4}}{8^{4}}=\dfrac{8^{7}}{8^{4}} \times \dfrac{5^{4}}{5^{7}}=8^{3} \times 5^{-3}=$ $512 \times \dfrac{1}{5^{3}}=512 \times \dfrac{1}{125}=4.096$

7. सरल कीजिए :

(i) $\quad \dfrac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}(t \neq 0)$

उत्तर: (i) $\dfrac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}=\dfrac{25 \times \dfrac{1}{t^{4}}}{\dfrac{1}{5^{3}} \times 10 \times \dfrac{1}{t^{8}}}=\dfrac{25 \times 5^{3} \times t^{8}}{\left(10 \times t^{4}\right)}=\dfrac{25 \times 125 \times t^{4}}{10}=\dfrac{3125 \times t^{4}}{10}$

(ii) $\dfrac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$

उत्तर: (ii) $5^{2}=5^{3} \times 5^{2}=5^{5^{3}+2^{6^{5}}}=5^{5}$

प्रश्नावली 12.2

1. निम्नलखित संख्याओं को मानक रूप में व्यक्त कीजिए।

(i) $0.000000000085$

उत्तर: $0.000000000085=8.5 \times 10^{-11}$

(ii) $0.0000000000942$

उत्तर: $0 .0000000000942=9.42 \times 10^{-11}$

(iii) 6020000000000000

उत्तर: $6.02 \times 10^{15}$

(iv) $0.00000000837$

उत्तर: $0.00000000837=8.37 \times 10^{-9}$

(v) 31860000000

उत्तर: $31860000000=3.186 \times 10^{10}$

2. निम्नलिखत संख्याओं को सामान्य रूप में व्यक्त कीजिए।

(i) $\quad 3.02 \times 10^{-6}$

उत्तर: $3.02 \times 10^{-6}=0.00000302$

(ii) $\quad 4.5 \times 10^{4}$

उत्तर: $4 .5 \times 10^{4}=45000$

(iii) $3 \times 10-8$

उत्तर: $3 \times 10^{-8}=0.00000003$

(iv) $\quad 1.0001 \times 10^{9}$

उत्तर: $ \times 10^{9}=1000100000$

(v) $\quad 5.8 \times 10^{12}$

उत्तर: $5 .8 \times 10^{12}=5800000000000$

(vi) $\quad 3.61492 \times 10^{6}$

उत्तर: $3 .61492 \times 10^{6}=3614920$

3. निम्लिखित कथनों में को संख्या प्रकट हो रही है, उन्हें मानक रूप में प्रकट कीजिए।

(i) 1 माइक्रोन $1 / 1000000 m$ के बराबर होता है।

उत्तर:  $1 / 1000000=1 \times 10^{-6}$

(ii) एक इलेक्ट्रॉन आवेश $0.000,000,000,000,000,00016$ कुलंब होता है।

उत्तर:   $\times 10-{ }^{19}$

(iii) जीवाणु की माप $0.0000005 m$ है।

उत्तर:  $0 .0000005=5 \times 10^{-7}$

(iv) पौधों की कोशिकाओं की माप $0.00001275 m$ है।

उत्तर:   $0.00001275=1.275 \times 10^{-5}$

(v) मोटे कागज की मोटाई $0.07 mm$ है।

उत्तर:  $0.07=7 \times 10^{-2}$

4. एक ढेर में पांच किताबें है, जिनमें प्रत्येक की मोटाई $20 mm$ तथा पांच कागज की पत्रक

है। जिनमें प्रत्येक की मोटाई $0.016 mm$ है। इस ढेर की कुल मोटाई ज्ञात कीजिए।

उत्तर:  दिया गया 

एक पुस्तक की मोटाई $=20 mm$

5 पुस्तकों की मोटाई $=(5 \times 20) mm =100 mm$

एक कागज की मोटाई = $0.016 mm$

इसलिए 5 कागजों की मोटाई = $0.016 \times 5=0.080 mm$

एक ढेर की कुल मोटाई

$=5$ पुस्तकों की मोटाई $+5$ कागजों की मोटाई

$=(100+0.08) mm$

$=100.08 mm$

$=1.0008 \times 10^{2} mm$

ढेर की कुल मोटाई$=1.0008 \times 10^{2} mm$

NCERT Solutions for Class 8 Maths Chapter 12 Exponents and Powers in Hindi

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