NCERT Solutions for Class 7 Maths In Hindi Chapter 13 Exponents and Powers

प्रश्नावली 13.1

1. निम्नलिखित के मान ज्ञात कीजिए:

1. \[{{\mathbf{2}}^{\mathbf{6}}}\]

उत्तर: \[2^{6}2^{6}=2\times 2\times 2\times 2\times 2\times 2=64\]

2. \[{{\mathbf{9}}^{\mathbf{3}}}\]

उत्तर: \[{{9}^{3}}=9\times 9\times 9=729\]

3. \[\mathbf{1}{{\mathbf{1}}^{\mathbf{2}}}\]

उत्तर: \[{{11}^{2}}=11\times 11=121\]

4. \[{{\mathbf{5}}^{\mathbf{4}}}\]

उत्तर: \[{{5}^{4}}=5\times 5\times 5\times 5=625\]

2. निम्नलिखित को घातांकीय रूप में व्यक्त कीजिए:

1. \[\mathbf{6}\times \mathbf{6}\times \mathbf{6}\times \mathbf{6}\]

2. \[\mathbf{t}\times \mathbf{t}\]

3. \[\mathbf{b}\times \mathbf{b}\times \mathbf{b}\times \mathbf{b}\]

4. \[\mathbf{5}\times \mathbf{5}\times \mathbf{7}\times \mathbf{7}\times \mathbf{7}\]

5. \[\mathbf{2}\times \mathbf{2}\times \mathbf{a}\times \mathbf{a}\]

6. \[\mathbf{a}\times \mathbf{a}\times \mathbf{a}\times \mathbf{c}\times \mathbf{c}\times \mathbf{c}\times \mathbf{c}\times \mathbf{d}\]

उत्तर: 

\[\begin{align} & \left( i \right)~{{6}^{4}} \\  & \left( ii \right)~{{t}^{2}} \\  & \left( iii \right)~{{b}^{4}} \\  & \left( iv \right)~{{5}^{2}}\times {{7}^{3}} \\  & \left( v \right)~{{2}^{2}}\times {{a}^{2}} \\  \end{align}\]

3. निम्नलिखित संख्याओं में से प्रत्येक को घातांकीय संकेतन में व्यक्त कीजिए:

1. 512

उत्तर: \[512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2={{2}^{9}}\]

2. \[\mathbf{343}\]

उत्तर: \[343=7\times 7\times 7={{7}^{3}}\]

3. \[\left( \mathbf{c} \right)\text{ }\mathbf{729}\]

उत्तर: \[729=3\times 3\times 3\times 3\times 3\times 3={{3}^{6}}\]

4. \[\left( \mathbf{d} \right)\text{ }\mathbf{3125}\]

उत्तर: \[3125=5\times 5\times 5\times 5\times 5={{5}^{5}}\]

4. निम्नलिखित में से प्रत्येक भाग में, जहाँ भी सम्भव हो, बड़ी संख्या को पहचानिए:

1. \[{{\mathbf{4}}^{\mathbf{3}}}~\]या \[{{\mathbf{3}}^{\mathbf{4}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{4}^{3}}=4\times 4\times 4=64  \\ {{3}^{4}}=3\times 3\times 3\times 3=81  \\ {{3}^{4}}>~{{4}^{3}}  \\ \end{array}\] 

2. \[{{\mathbf{5}}^{\mathbf{3}}}~\]या \[{{\mathbf{3}}^{\mathbf{5}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{5}^{3}}=5\times 5\times 5=125  \\ {{3}^{5}}=3\times 3\times 3\times 3\times 3=243  \\ {{3}^{5}}>{{5}^{3}}  \\ \end{array}\] 

3. \[{{\mathbf{2}}^{\mathbf{8}}}~\]या \[{{\mathbf{8}}^{\mathbf{2}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{2}^{8}}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256  \\ {{8}^{2}}=8\times 8=64  \\ {{2}^{8}}>{{8}^{2}}  \\ \end{array}\] 

4. $\mathbf{100^{2}}$ या $\mathbf{2^{100}}$

उत्तर: 

\[\begin{array}{*{35}{l}} {{100}^{2}}={{\left( 2\times 2\times 5\times 5 \right)}^{2}}  \\ =\left( {{2}^{2}}\times {{5}^{2}} \right)2  \\ ={{2}^{4}}\times {{5}^{4}}  \\ \end{array}\]

इस गुणनखंड में \[5\text{ }>\text{ }4\]और \[4={{2}^{2}}\]

यदि यहाँ पर \[5\] के स्थान पर \[8\] भी होता तो \[8={{2}^{3}}\]`

\[5\]के स्थान पर \[8\] होने की स्थिति में गुणनखंड इस प्रकार होता:

\[{{2}^{4}}\times {{2}^{3}}={{2}^{7}}~\] जो हर हाल में \[{{2}^{100}}\] से छोटा होता।

इसलिए, \[{{2}^{100}}>{{100}^{2}}\]

5. \[{{\mathbf{2}}^{\mathbf{10}}}~\]या \[\mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{2}^{10}}={{4}^{5}}=4\times 4\times 4\times 4\times 4=1024  \\ {{10}^{2}}=100  \\ {{2}^{10}}>{{10}^{2}}  \\ \end{array}\]

5. निम्नलिखित में से प्रत्येक को उनके अभाज्य गुणनखंडों की घातो के गुणनफल के रूप में व्यक्त कीजिए।

1. 648

उत्तर: 

\[\begin{array}{*{35}{l}} 648=2\times 2\times 2\times 3\times 3\times 3\times 3  \\ ={{2}^{3}}\times {{3}^{4}}  \\ \end{array}\] 

2. \[\mathbf{405}\]

उत्तर:

 \[\begin{align} & 405 \\  & =5\times 3\times 3\times 3\times 3 \\  & =5\times {{3}^{4}} \\  \end{align}\]

3. \[\mathbf{540}\]

उत्तर: 

\[\begin{align} & 540=2\times 2\times 3\times 3\times 3\times 5 \\  & ={{2}^{2}}\times {{3}^{3}}\times 5 \\  \end{align}\] 

4. \[\mathbf{3600}\]

उत्तर: 

\[\begin{array}{*{35}{l}} 3600=36\times 100  \\ =4\times 9\times 4\times 25  \\ =4\times 4\times 9\times 5\times 5  \\ ={{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}  \\ \end{array}\]

6.सरल कीजिए:

1. \[\left( \mathbf{a} \right)\text{ }\mathbf{2}\text{ }\times \mathbf{1}{{\mathbf{0}}^{\mathbf{3}}}\]

उत्तर: \[2\times {{10}^{3}}=2\times 1000=2000\]

2. \[{{\mathbf{7}}^{\mathbf{2}}}~\times \text{ }{{\mathbf{2}}^{\mathbf{2}}}\]

उत्तर: \[{{7}^{2}}\times {{2}^{2}}=49\times 4=196\]

3. \[{{\mathbf{2}}^{\mathbf{3}}}~\times \text{ }\mathbf{5}\]

उत्तर: \[{{2}^{3}}\times 5=8\times 5=40\]

4. \[\mathbf{3}\text{ }\times \text{ }{{\mathbf{4}}^{\mathbf{4}}}\]

उत्तर: \[3\times {{4}^{4}}=3\times 256=768\]

5. \[\mathbf{0}\text{ }\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}\]

उत्तर: \[0\times 102=0\]

6. \[{{\mathbf{5}}^{\mathbf{2}}}~\times \text{ }{{\mathbf{3}}^{\mathbf{3}}}\]

उत्तर: \[{{5}^{2}}\times {{3}^{3}}=25\times 27=675\]

7. \[{{\mathbf{2}}^{\mathbf{4}~}}\times \text{ }{{\mathbf{3}}^{\mathbf{2}}}\]

उत्तर: \[{{2}^{4}}\times {{3}^{2}}=16\times 9=144\]

8. \[{{\mathbf{3}}^{\mathbf{2}}}~\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{4}}}\]

उत्तर:  \[{{3}^{2}}\times {{10}^{4}}=9\times 10000=90000\]

7. सरल कीजिए:

(a) \[{{\left( -\mathbf{4} \right)}^{\mathbf{3}}}\]

उत्तर: \[{{\left( -4 \right)}^{3}}=-64\]

\[\left( \mathbf{b} \right)\text{ }\left( -\mathbf{3} \right)\text{ }\times \text{ }{{\left( -\mathbf{2} \right)}^{\mathbf{3}}}\]

उत्तर: \[\left( -3 \right)\times {{\left( -2 \right)}^{3}}=\left( -3 \right)\times \left( -8 \right)=24\]

\[\left( \mathbf{c} \right)\text{ }{{\left( -\mathbf{3} \right)}^{\mathbf{2}}}~\times \text{ }{{\left( -\mathbf{5} \right)}^{\mathbf{2}}}\]

उत्तर: \[{{\left( -3 \right)}^{2}}\times {{\left( -5 \right)}^{2}}=9\times 25=225\]

\[\left( \mathbf{d} \right)\text{ }{{\left( -\mathbf{2} \right)}^{\mathbf{3}}}~\times \text{ }{{\left( -\mathbf{10} \right)}^{\mathbf{3}}}\]

उत्तर: \[{{\left( -2 \right)}^{3}}\times {{\left( -10 \right)}^{3}}=\left( -8 \right)\times \left( -1000 \right)=8000\]

8. निम्नलिखित संख्याओं की तुलना कीजिए:

\[\left( \mathbf{a} \right)\text{ }\mathbf{2}.\mathbf{7}\text{ }\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{12}}},\text{ }\mathbf{1}.\mathbf{5}\text{ }\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{8}}}\]

उत्तर:

 \[2.7 > 1.5\]

\[\begin{array}{*{35}{l}} {{10}^{12}}>{{10}^{8}}  \\ 2.7\times {{10}^{12}}>1.5\times {{10}^{8}}  \\ \end{array}\] 

\[\left( \mathbf{b} \right)\text{ }\mathbf{4}\text{ }\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{14}}},\text{ }\mathbf{3}\text{ }\times \text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{17}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{10}^{14}}<{{10}^{17}}  \\ 4\times {{10}^{14}}<3\times {{10}^{17}}  \\ \end{array}\]

प्रश्नावली13.2

1. घातांकों के नियमों का प्रयोग करते हुए, सरल कीजिए और उत्तर को घातांकीय रूप में लिखिए:

1. \[{{\mathbf{3}}^{\mathbf{2}}}\times {{\mathbf{3}}^{\mathbf{4}}}\times {{\mathbf{3}}^{\mathbf{8}}}\]

उत्तर:

\[\begin{array}{*{35}{l}} ~{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}  \\ {{3}^{2}}\times {{3}^{4}}\times {{3}^{8}}={{3}^{2+4+8}}  \\ ={{3}^{14}}  \\ \end{array}\] 

2. \[{{\mathbf{6}}^{\mathbf{15}}}\div {{\mathbf{6}}^{\mathbf{10}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}  \\ {{6}^{15}}\div {{6}^{10}}={{6}^{15-10}}  \\ ={{6}^{5}}  \\ \end{array}\] 

3. \[{{\mathbf{a}}^{\mathbf{3}}}\times {{\mathbf{a}}^{\mathbf{2}}}\]

उत्तर: 

\[{{a}^{3}}\times {{a}^{2}}={{a}^{3+2}}={{a}^{5}}\] 

4. \[{{\mathbf{7}}^{\mathbf{x}}}\times {{\mathbf{7}}^{\mathbf{2}}}\]

उत्तर: 

\[{{7}^{x}}\times {{7}^{2}}={{7}^{x+2}}\] 

5. \[{{\left( \mathbf{5^2} \right)}^{\mathbf{3}}}\div {{\mathbf{5}}^{\mathbf{3}}}\]

उत्तर:

\[\begin{array}{*{35}{l}} ~{{\left( {{5}^{2}} \right)}^{3}}\div {{5}^{3}}  \\ =\dfrac{{{5}^{2+3}}}{{{5}^{3}}}  \\ =\dfrac{{{5}^{5}}}{{{5}^{3}}}  \\ \begin{align} & ={{5}^{5-3}} \\  & ={{5}^{2}} \\  \end{align}  \\ \end{array}\] 

6. \[{{\mathbf{2}}^{\mathbf{5}}}\times {{\mathbf{5}}^{\mathbf{5}}}\]

उत्तर: 

\[{{2}^{5}}\times {{5}^{5}}={{10}^{5}}\] 

7. \[{{\mathbf{a}}^{\mathbf{4}}}\times {{\mathbf{b}}^{\mathbf{4}}}\]

उत्तर: 

\[{{a}^{4}}\times {{b}^{4}}={{\left( ab \right)}^{4}}\] 

8. \[{{\left( {{\mathbf{3}}^{\mathbf{4}}} \right)}^{\mathbf{5}}}\]

उत्तर:

 \[{{\left( {{3}^{4}} \right)}^{5}}={{3}^{4\times 5}}={{3}^{20}}\]

9. \[\left( {{\mathbf{2}}^{\mathbf{20}}}\div {{\mathbf{2}}^{\mathbf{15}}} \right)\times {{\mathbf{2}}^{\mathbf{3}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \left( \dfrac{{{2}^{20}}}{{{2}^{15}}} \right)\times {{2}^{3}}  \\ =\left( {{2}^{20-15}} \right)\times {{2}^{3}}  \\ \begin{align} & ={{2}^{5}}\times {{2}^{3}} \\  & ={{2}^{5+3}} \\  & ={{2}^{8}} \\  \end{align}  \\ \end{array}\] 

10. \[{{\mathbf{8}}^{\mathbf{t}}}\div {{\mathbf{8}}^{\mathbf{2}}}\]

उत्तर: 

\[\dfrac{{{8}^{t}}}{{{8}^{2}}}={{8}^{t-2}}\]

2. निम्नलिखित में से प्रत्येक को सरल करके घातांकीय रूप में व्यक्त कीजिए:

1. \[\dfrac{{{\mathbf{2}}^{\mathbf{3}}}\times {{\mathbf{3}}^{\mathbf{4}}}\times \mathbf{4}}{\mathbf{3}\times \mathbf{32}}\]

उत्तर: 

$\dfrac{{{2}^{3}}\times {{3}^{4}}\times 4}{3\times 32} $

$ =\dfrac{{{2}^{3}}\times {{3}^{4}}\times {{2}^{2}}}{3\times {{2}^{5}}}  $

$ ={{2}^{3+2-5}}\times {{3}^{4-1}} $

$ ={{2}^{0}}\times {{3}^{3}} $

$ =1\times {{3}^{3}} $

$={{3}^{3}} $

2. \[\left( {{\left( {{\mathbf{5}}^{\mathbf{2}}} \right)}^{\mathbf{3}}}\times {{\mathbf{5}}^{\mathbf{4}}} \right)\div {{\mathbf{5}}^{\mathbf{7}}}\]

उत्तर:

\[\dfrac{\left( {{\left( {{5}^{2}} \right)}^{3}}\times {{5}^{4}} \right)}{{{5}^{7}}}\]

$   =\dfrac{{{5}^{6}}\times {{5}^{4}}}{{{5}^{7}}}$

$5^{6+4-7}$

$ ={{5}^{3}} $

3. \[\mathbf{2}{{\mathbf{5}}^{\mathbf{4}}}\div {{\mathbf{5}}^{\mathbf{3}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{{{25}^{4}}}{{{5}^{3}}}  \\ =\dfrac{{{\left( {{5}^{2}} \right)}^{4}}}{{{5}^{3}}}  \\ =\dfrac{{{5}^{8}}}{{{5}^{3}}}  \\ \begin{align} & ={{5}^{8-3}} \\  & ={{5}^{5}} \\  \end{align}  \\ \end{array}\] 

4. \[\dfrac{\mathbf{3}\times {{\mathbf{7}}^{\mathbf{2}}}\times \mathbf{1}{{\mathbf{1}}^{\mathbf{8}}}}{\mathbf{21}\times \mathbf{1}{{\mathbf{1}}^{\mathbf{3}}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{3\times {{7}^{2}}\times {{11}^{8}}}{21\times {{11}^{3}}}  \\ =\dfrac{3\times {{7}^{2}}\times {{11}^{8}}}{3\times 7\times {{11}^{3}}}  \\ ={{3}^{1-1}}\times {{7}^{2-1}}\times {{11}^{8-3}}  \\ \begin{align} & ={{3}^{0}}\times {{7}^{1}}\times {{11}^{5}} \\  & =7\times {{11}^{5}} \\ \end{align}  \\ \end{array}\] 

5. \[\dfrac{{{\mathbf{3}}^{\mathbf{7}}}}{{{\mathbf{3}}^{\mathbf{4}}}\times {{\mathbf{3}}^{\mathbf{3}}}}\]

उत्तर: 

\[\dfrac{{{3}^{7}}}{{{3}^{4}}\times {{3}^{3}}}\]

\[\begin{array}{*{35}{l}} =\dfrac{{{3}^{7}}}{{{3}^{4+3}}}=\dfrac{{{3}^{7}}}{{{3}^{7}}}  \\ ={{3}^{7-7}}={{3}^{0}}=1  \\ \end{array}\] 

6. \[{{\mathbf{2}}^{\mathbf{0}}}+{{\mathbf{3}}^{\mathbf{0}}}+{{\mathbf{4}}^{\mathbf{0}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{2}^{0}}+{{3}^{0}}+{{4}^{0}}  \\ \begin{align} & =1+1+1 \\  & =3 \\  \end{align}  \\ \end{array}\] 

7. \[{{\mathbf{2}}^{\mathbf{0}}}\times {{\mathbf{3}}^{\mathbf{0}}}\times {{\mathbf{4}}^{\mathbf{0}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{2}^{0}}\times {{3}^{0}}\times {{4}^{0}}  \\ \begin{align} & =1\times 1\times 1 \\  & =1 \\  \end{align}  \\ \end{array}\] 

8. \[\left( {{\mathbf{3}}^{\mathbf{0}}}+{{\mathbf{2}}^{\mathbf{0}}} \right)\times {{\mathbf{5}}^{\mathbf{0}}}\]

उत्तर:

\[\begin{array}{*{35}{l}} ~\left( {{3}^{0}}+{{2}^{0}} \right)\times {{5}^{0}}  \\ \begin{align} & =\left( 1+1 \right)\times 1 \\  & =2\times 1 \\  & =2 \\ \end{align}  \\ \end{array}\] 

9. \[\dfrac{{{\mathbf{2}}^{\mathbf{8}}}\times {{\mathbf{a}}^{\mathbf{5}}}}{{{\mathbf{4}}^{\mathbf{3}}}\times {{\mathbf{a}}^{\mathbf{3}}}}\]

उत्तर:

\[\begin{array}{*{35}{l}} ~\dfrac{{{2}^{8}}\times {{a}^{5}}}{{{4}^{3}}\times {{a}^{3}}}  \\ =\dfrac{{{2}^{8}}\times {{a}^{5}}^{-3}}{{{2}^{6}}}  \\ ={{2}^{8-6}}\times {{a}^{2}}={{2}^{2}}\times {{a}^{2}}={{\left( 2a \right)}^{2}}  \\ \end{array}\] 

10. \[\dfrac{~{{\mathbf{a}}^{\mathbf{5}}}\times {{\mathbf{a}}^{\mathbf{3}}}}{{{\mathbf{a}}^{\mathbf{8}}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{{{a}^{5}}\times {{a}^{3}}}{{{a}^{8}}}  \\ \begin{align} & ={{a}^{5+3-8}} \\  & ={{a}^{0}} \\  & =1 \\  \end{align}  \\ \end{array}\] 

11. \[\dfrac{{{\mathbf{4}}^{\mathbf{5}}}\times {{\mathbf{a}}^{\mathbf{8}}}{{\mathbf{b}}^{\mathbf{3}}}}{{{\mathbf{4}}^{\mathbf{5}}}\times {{\mathbf{a}}^{\mathbf{5}}}{{\mathbf{b}}^{\mathbf{2}}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{{{4}^{5}}\times {{a}^{8}}{{b}^{3}}}{{{4}^{5}}\times {{a}^{5}}{{b}^{2}}}  \\ =\dfrac{{{a}^{8}}{{b}^{3}}}{{{a}^{5}}{{b}^{2}}}  \\ \begin{align} & ={{a}^{8-5}}\times {{b}^{3-2}} \\  & ={{a}^{3}}\times b \\ \end{align}  \\ \end{array}\] 

12. \[{{\left( {{\mathbf{2}}^{\mathbf{3}}}\times \mathbf{2} \right)}^{\mathbf{2}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} {{\left( {{2}^{3}}\times 2 \right)}^{2}}  \\ \begin{align} & ={{\left( {{2}^{3+1}} \right)}^{2}} \\  & ={{\left( {{2}^{4}} \right)}^{2}} \\  & ={{2}^{8}} \\  \end{align}  \\ \end{array}\]

3. बताइए कि निम्नलिखित कथन सत्य है या असत्य तथा अपने उत्तर का कारण भी दीजिए:

1. \[\mathbf{10}\times\mathbf{1}{{\mathbf{0}}^{\mathbf{11}}}=\mathbf{10}{{\mathbf{0}}^{\mathbf{11}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} LHS\text{ }=~10\times {{10}^{11}}={{10}^{1+11}}={{10}^{12}}  \\ \begin{align} & RHS\text{ }=~{{100}^{11}}={{\left( {{10}^{2}} \right)}^{11}}={{10}^{22}} \\  & LHS\ne RHS \\ \end{align}  \\ {}  \\ \end{array}\] 

अतः कथन असत्य है।

2. \[{{\mathbf{2}}^{\mathbf{3}}}>{{\mathbf{5}}^{\mathbf{2}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} LHS\text{ }=~{{2}^{3}}=8  \\ RHS\text{ }=~{{5}^{2}}=25  \\ LHS\ne RHS  \\ \end{array}\] 

अतः कथन असत्य है।

3.  $\mathbf{2^3×3^2=6^5}$

उत्तर: 

\[\begin{array}{*{35}{l}} LHS\text{ }=~{{2}^{3}}\times {{3}^{2}}=8\times 9=72  \\ RHS\text{ }=~{{6}^{5}}=7776  \\ LHS\ne RHS  \\ \end{array}\] 

4. \[{{\mathbf{3}}^{\mathbf{0}}}=\mathbf{100}{{\mathbf{0}}^{\mathbf{0}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} LHS\text{ }=~{{3}^{0}}=1  \\ RHS\text{ }=~{{1000}^{0}}=1  \\ LHS=RHS  \\ \end{array}\]

अतः कथन सत्य है।

4. निम्नलिखित में से प्रत्येक को केवल अभाज्य गुणनखंडों की घातों के गुणनफल के रूप में व्यक्त कीजिए:

1. \[\mathbf{108}\text{ }\times \text{ }\mathbf{192}\]

उत्तर:

 \[\begin{array}{*{35}{l}} 108\times 192  \\ ={{2}^{2}}\times {{3}^{3}}\times 192  \\ =2^2\times 3^3\times 2^6\times 3=2^2\times 3^3\times 2^6\times 3  \\ ={{2}^{2+6}}\times {{3}^{3+1}}={{2}^{8}}\times {{3}^{4}}  \\ \end{array}\] 

2. \[\mathbf{270}\]

उत्तर: ल0स0 लेने पर ,

\[270=2\times {{3}^{3}}\times 5\] 

3. \[\text{ }\mathbf{729}\text{ }\times \text{ }\mathbf{64}\]

उत्तर: ल0स0 लेने पर

 \[\begin{array}{*{35}{l}} 729\times 64  \\ =9\times 9\times 9\times 8\times 8  \\ ={{3}^{2}}\times {{3}^{2}}\times {{3}^{2}}\times {{2}^{3}}\times {{2}^{3}}  \\ ={{3}^{2+2+2}}\times {2^{ 3+3}}  \\ =3^6\times 2^6=3^6\times 2^6  \\ \end{array}\] 

4. \[\text{ }\mathbf{768}\]

उत्तर: ल0स0 लेने पर

\[~768={{2}^{8}}\times 3\]

5. सरल कीजिए:

1. \[\dfrac{\left( {{\mathbf{2}}^{\mathbf{5}}} \right)\mathbf{2}\times {{\mathbf{7}}^{\mathbf{3}}}}{{{\mathbf{8}}^{\mathbf{3}}}\times \mathbf{7}}\]

उत्तर: \[\dfrac{{{\left( {{2}^{5}} \right)}^{2}}\times 7}{{{8}^{3}}\times 7^3}\]

\[\begin{array}{*{35}{l}} {}  \\ =\dfrac{{{2}^{10}}\times {{7}^{3}}}{{{2}^{9}}\times 7}  \\ \end{array}\] \[\begin{array}{*{35}{l}} \begin{align} & ={{2}^{10-9}}\times {{7}^{3-1}} \\  & ={{2}^{1}}\times {{7}^{2}} \\  \end{align}  \\ {}  \\ \end{array}\] 

2. \[\dfrac{~\mathbf{25}\times {{\mathbf{5}}^{\mathbf{2}}}\times {{\mathbf{t}}^{\mathbf{8}}}}{\mathbf{1}{{\mathbf{0}}^{\mathbf{3}}}\times {{\mathbf{t}}^{\mathbf{4}}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{25\times {{5}^{2}}\times {{t}^{8}}}{{{10}^{3}}\times {{t}^{4}}}  \\ =\dfrac{{{5}^{2}}\times {{5}^{2}}\times {{t}^{8}}}{{{2}^{3}}\times {{5}^{3}}\times {{t}^{4}}}  \\ =\left( {{5}^{4-3}} \right)\times \dfrac{{{t}^{8-4}}}{{{2}^{3}}}  \\ =\dfrac{5\times {{t}^{4}}}{{{2}^{3}}} \\ \end{array}\] 

3. \[\dfrac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}\]

उत्तर: 

\[\begin{array}{*{35}{l}} \dfrac{{{3}^{5}}\times {{10}^{5}}\times 25}{{{5}^{7}}\times {{6}^{5}}}  \\ =\dfrac{{{3}^{5}}\times {{2}^{5}}\times {{5}^{5}}\times {{5}^{2}}}{{{5}^{7}}\times {{2}^{5}}\times {{3}^{5}}}  \\ ={{3}^{5-5}}\times {{2}^{5-5}}\times {{5}^{5+2-7}}  \\ \begin{align} & ={{3}^{0}}\times {{2}^{0}}\times {{5}^{0}} \\  & =1\times 1\times 1 \\  & =1 \\ \end{align}  \\ \end{array}\]

प्रश्नावली 13.3

1. निम्नलिखित संख्याओं को प्रसारित रूप में लिखिए:

1. \[\text{ }\mathbf{279404}\]

उत्तर: \[2\text{ }\times \text{ }{{10}^{5}}~+\text{ }7\text{ }\times \text{ }{{10}^{4~}}+\text{ }9\text{ }\times \text{ }{{10}^{3}}~+\text{ }4\text{ }\times \text{ }{{10}^{2}}~+\text{ }4\text{ }\times \text{ }{{10}^{0}}\]

2. \[\text{ }\mathbf{3006194}\]

उत्तर: \[3\text{ }\times \text{ }{{10}^{6~}}+\text{ }6\text{ }\times \text{ }{{10}^{3}}~+\text{ }1\text{ }\times \text{ }{{10}^{2}}~+\text{ }9\text{ }\times \text{ }{{10}^{1}}~+\text{ }4\text{ }\times \text{ }{{10}^{0}}\]

3. \[\text{ }\mathbf{2806196}\]

उत्तर: \[2\text{ }\times \text{ }{{10}^{6}}~+\text{ }8\text{ }\times \text{ }{{10}^{5}}~+\text{ }6\text{ }\times \text{ }{{10}^{3}}~+\text{ }1\text{ }\times \text{ }{{10}^{2}}~+\text{ }9\text{ }\times \text{ }{{10}^{1}}~+\text{ }6\text{ }\times \text{ }{{10}^{0}}\]

4. \[\text{ }\mathbf{120719}\]

उत्तर: \[1\text{ }\times \text{ }{{10}^{5~}}+\text{ }2\text{ }\times \text{ }{{10}^{4}}~+\text{ }7\text{ }\times \text{ }{{10}^{2~}}+\text{ }1\text{ }\times \text{ }{{10}^{1}}~+9\text{ }\times \text{ }{{10}^{0}}\]

5. \[\text{ }\mathbf{20068}\]

उत्तर: \[2\text{ }\times \text{ }{{10}^{4}}~+\text{ }6\text{ }\times \text{ }{{10}^{1}}~+\text{ }8\text{ }\times \text{ }{{10}^{0}}\]

2. निम्नलिखित प्रसारित रूपों में से प्रत्येक के लिए संख्या ज्ञात कीजिए:

1. \[\mathbf{8}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{4}}}+\mathbf{6}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{3}}}+\mathbf{0}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}+\mathbf{4}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{1}}}+\mathbf{5}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{0}}}\]

2. \[\mathbf{4}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{5}}}+\mathbf{5}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{3}}}+\mathbf{3}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}+\mathbf{2}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{0}}}\]

3. \[\mathbf{3}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{4}}}+\mathbf{7}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}+\mathbf{5}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{0}}}\]

4. \[\mathbf{9}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{5}}}+\mathbf{2}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}+\mathbf{3}\times \mathbf{1}{{\mathbf{0}}^{\mathbf{1}}}\]

उत्तर: \[\left( i \right)\text{ }86045,\text{ }\left( ii \right)\text{ }400532,\text{ }\left( iii \right)\text{ }30705,\text{ }\left( iv \right)\text{ }90023\]

3. निम्नलिखित संख्याओं को मानक रूप में व्यक्त कीजिए:

1. \[\text{ }\mathbf{5},\mathbf{00},\mathbf{00},\mathbf{000}\]

उत्तर: \[~5\text{ }\times \text{ }{{10}^{7}}\]

2. \[\text{ }\mathbf{70},\mathbf{00},\mathbf{000}\]

उत्तर: \[7\text{ }\times \text{ }{{10}^{6}}\]

3. 3, 18, 65, 00, 00

उत्तर: \[3.1865\text{ }\times \text{ }{{10}^{9}}\]

4. \[\text{ }\mathbf{3},\mathbf{90},\mathbf{878}\]

उत्तर: \[3.90878\text{ }\times \text{ }{{10}^{5}}\]

5. \[\text{ }\mathbf{39087}.\mathbf{8}\]

उत्तर: \[3.90878\text{ }\times \text{ }{{10}^{4}}\]

6. \[\text{ }\mathbf{3908}.\mathbf{78}\]

उत्तर: \[3.90878\text{ }\times \text{ }{{10}^{3}}\]

4. निम्नलिखित प्रश्नों में प्रकट होने वाली (आने वाली) संख्याओं को मानक रूप में व्यक्त कीजिए।

1. पृथ्वी और चंद्रमा के बीच की दूरी \[\mathbf{384},\mathbf{000},\mathbf{000}\text{ }\mathbf{m}\] है।

उत्तर: \[3.84\text{ }\times \text{ }{{10}^{8}}~m\]

2. निर्वात स्थान में प्रकाश की चाल (या वेग) \[\mathbf{300},\mathbf{000},\mathbf{000}\left( \text{ }\mathbf{m} \right)/\left( \mathbf{sec}. \right)\]है।

उत्तर: \[3\text{ }\times \text{ }{{10}^{8}}~m/sec\]

3. पृथ्वी का व्यास \[\mathbf{12756000}\text{ }\mathbf{m}\] है।

उत्तर: \[1.2756\text{ }\times \text{ }{{10}^{7}}\]

4. सूर्य का व्यास \[\mathbf{1},\mathbf{400},\mathbf{000},\mathbf{000}\text{ }\mathbf{m}\] है।

उत्तर: \[1.4\text{ }\times \text{ }{{10}^{9}}~m\]

5. एक आकाश गंगा में औसतन \[\mathbf{100},\mathbf{000},\mathbf{000},\mathbf{000}\]तारे हैं।

उत्तर: \[1\text{ }\times \text{ }{{10}^{11}}\]

6. विश्व मंडलमंडल (या सौर मंडल) \[\mathbf{12},\mathbf{000},\mathbf{000},\mathbf{000}\] वर्ष पुराना आकलित किया गया है।

उत्तर: \[1.2\text{ }\times \text{ }{{10}^{10}}~years\]

7. आकाश गंगा के मध्य से सूर्य की दूरी \[\mathbf{300},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000}\text{ }\mathbf{m}\] आकलित की गई है।

उत्तर: \[3\text{ }\times \text{ }{{10}^{20}}~m\]

8. \[\mathbf{1}.\mathbf{8}\text{ }\mathbf{g}\] भार वाली पानी की एक बूंद में \[\mathbf{60},\mathbf{230},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000},\mathbf{000}\]अणु (molecules) होते हैं।

उत्तर: \[6.023\text{ }\times \text{ }{{10}^{22~}}molecules\]

9. पृथ्वी में \[1,353,000,000k{{m}^{3}}\]समुद्र जल है।

उत्तर: \[1.353\text{ }\times \text{ }{{10}^{9}}~k{{m}^{3}}\]

10. मार्च 2001 में भारत की जनसंख्या \[\mathbf{1},\mathbf{027},\mathbf{000},\mathbf{000}\]थी।

उत्तर: \[1.027\text{ }\times \text{ }{{10}^{9}}\]

NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers In Hindi

Chapter-wise NCERT Solutions are provided everywhere on the internet with an aim to help the students to gain a comprehensive understanding. Class 7 Maths Chapter 13 solution Hindi mediums are created by our in-house experts keeping the understanding ability of all types of candidates in mind. NCERT textbooks and solutions are built to give a strong foundation to every concept. These NCERT Solutions for Class 7 Maths Chapter 13 in Hindi ensure a smooth understanding of all the concepts including the advanced concepts covered in the textbook.

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NCERT Solutions in Hindi medium have been created keeping those students in mind who are studying in a Hindi medium school. These NCERT Solutions for Class 7 Maths Exponents and Powers in Hindi medium pdf download have innumerable benefits as these are created in simple and easy-to-understand language. The best feature of these solutions is a free download option. Students of Class 7 can download these solutions at any time as per their convenience for self-study purpose. 

These solutions are nothing but a compilation of all the answers to the questions of the textbook exercises. The answers/ solutions are given in a stepwise format and very well researched by the subject matter experts who have relevant experience in this field. Relevant diagrams, graphs, illustrations are provided along with the answers wherever required. In nutshell, NCERT Solutions for Class 7 Maths in Hindi come really handy in exam preparation and quick revision as well prior to the final examinations.