NCERT Solutions Class 8 Maths Chapter 10 Exponents and Powers Exercise 10.1

NCERT Solutions Class 8 Maths Chapter 10 Exponents and Powers Exercise 10.1 introduces students to the concept of expressing large numbers in a shorter form using exponents. This exercise helps in understanding how to write numbers like 1,000,000 or 0.0001 using powers of 10, making calculations easier and faster. It also explains the rules of multiplying and dividing numbers with the same base.

These NCERT Solutions are designed as per the official NCERT textbook and explained in simple steps to help students grasp the topic easily. With clear methods and solved examples, students can practice and build confidence in using exponents. Solving NCERT Solutions from Exercise 10.1 helps Class 8 students develop strong foundational skills for advanced mathematics.

1.0Download NCERT Solutions Class 8 Maths Chapter 10 Exponents and Powers Exercise 10.1: Free PDF

Download the free NCERT Solutions Class 8 Maths Chapter 10 Exponents and Powers Exercise 10.1 PDF with easy-to-follow answers to help you practice and score well in exams.

2.0Key Concepts in Exercise 10.1 of Class 8 Maths Chapter 10

Exercise 10.1 of Chapter 10 – Exponents and Powers introduces the foundational concept of exponents, making it easier for students to work with very large or very small numbers in a compact form. It builds the base for efficient calculations using powers.

• Understanding Exponents
  Students learn how to represent repeated multiplication of the same number using exponents.
  Example:
• Base and Exponent
  The exercise helps students identify the base and exponent in expressions like anan, where:
• • a is the base
  • n is the exponent or power
• Laws of Exponents (Introductory)
  Basic rules are introduced, such as:
• • Multiplying powers with the same base:
  • Dividing powers with the same base:
  • Power of a power:
• Positive Integral Exponents
  Students practice working with exponents where both base and exponent are positive integers.
• Simple Calculations and Patterns
  The exercise includes evaluating and simplifying expressions involving powers to observe patterns and develop familiarity.

3.0NCERT Class 8 Maths Chapter 10: Other Exercises

4.0NCERT Class 8 Maths Chapter 10 Exercise 10.1 : Detailed Solutions

1. Evaluate. (i) $3^{-2}$ (ii) $(-4{)}^{-2}$ (iii) ${\left(\frac{1}{2}\right)}^{-5}$ Sol. (i) $3^{-2}=\frac{1}{3^2}=\frac{1}{3\times 3}=\frac{1}{9}$ (ii) $(-4{)}^{-2}=\frac{1}{(-4{)}^2}=\frac{1}{(-1{)}^2\times (4{)}^2}=\frac{1}{1\times 16}=\frac{1}{16}$ (iii) ${\left(\frac{1}{2}\right)}^{-5}=2^5=2\times 2\times 2\times 2\times 2=32$
2. Simplify and express the result in power notation with positive exponent. (i) $(-4{)}^5÷(-4{)}^8$ (ii) ${\left(\frac{1}{2^3}\right)}^2$ (iv) $\left(3^{-7}÷3^{-10}\right)\times 3^{-5}=\frac{3^{-7}}{3^{-10}}\times 3^{-5}$ $=3^{-7+10}\times 3^{-5}=3^3\times 3^{-5}$ $=3^{3+(-5)}=3^{3-5}=3^{-2}=\frac{1}{3^2}$ (v) $2^{-3}\times (-7{)}^{-3}$ $=-\frac{1}{2^3}\times \frac{1}{7^3}=-\frac{1}{(2\times 7{)}^3}=-\frac{1}{14^3}$
3. Find the value of (i) $\left(3^0+4^{-1}\right)\times 2^2$ (ii) $\left(2^{-1}\times 4^{-1}\right)÷2^{-2}$ (iii) ${\left(\frac{1}{2}\right)}^{-2}+{\left(\frac{1}{3}\right)}^{-2}+{\left(\frac{1}{4}\right)}^{-2}$ (iv) ${\left(3^{-1}+4^{-1}+5^{-1}\right)}^0$ (v) ${\left\{{\left(\frac{-2}{3}\right)}^{-2}\right\}}^2$ Sol. (i) $\left(3^0+4^{-1}\right)\times 2^2=\left(1+\frac{1}{4}\right)\times 4$ $=\frac{4+1}{4}\times 4=\frac{5}{4}\times 4=5$ (ii) $\left(2^{-1}\times 4^{-1}\right)÷2^{-2}=\left(\frac{1}{2}\times \frac{1}{4}\right)÷\frac{1}{2^2}=\frac{1}{8}÷\frac{1}{4}$ $=\frac{1}{8}\times \frac{4}{1}=\frac{1}{2}$ (iii) ${\left(\frac{1}{2}\right)}^{-2}+{\left(\frac{1}{3}\right)}^{-2}+{\left(\frac{1}{4}\right)}^{-2}=2^2+3^2+4^2$ $=4+9+16=29$ (iv) ${\left(3^{-1}+4^{-1}+5^{-1}\right)}^0$ $=1\left[\because a^0=1\right]$ (v) ${\left\{{\left(-\frac{2}{3}\right)}^{-2}\right\}}^2={\left\{{\left(-\frac{3}{2}\right)}^2\right\}}^2$ $={\left(-\frac{3}{2}\right)}^4=\frac{3^4}{2^4}=\frac{81}{16}$
4. Evaluate: (i) $\frac{8^{-1}\times 5^3}{2^{-4}}$ (ii) $\left(5^{-1}\times 2^{-1}\right)\times 6^{-1}$ Sol. (i) $\frac{8^{-1}\times 5^3}{2^{-4}}=\frac{1}{8}\times 5^3\times 2^4=\frac{1}{8}\times 125\times 16$ $=125\times 2=250$ (ii) $\left(5^{-1}\times 2^{-1}\right)\times 6^{-1}$ $=\left(\frac{1}{5}\times \frac{1}{2}\right)\times \frac{1}{6}=\frac{1}{10}\times \frac{1}{6}=\frac{1}{60}$
5. Find the value of $m$ for which $5^m÷5^{-3}=5^5$ Sol. We have, $5^m÷5^{-3}=5^5$ $\Rightarrow 5^{\mathrm{m}}÷\frac{1}{5^3}=5^5\Rightarrow 5^{\mathrm{m}}\times 5^3=5^5$ $\Rightarrow 5^{m+3}=5^5$ $\Rightarrow \mathrm{m}+3=5[\because$ if ${\mathrm{a}}^{\mathrm{m}}={\mathrm{a}}^{\mathrm{n}}$, then $\mathrm{m}=\mathrm{n}]$ $\Rightarrow m=5-3=2$
6. Evaluate (i) ${\left\{{\left(\frac{1}{3}\right)}^{-1}-{\left(\frac{1}{4}\right)}^{-1}\right\}}^{-1}$ (ii) ${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-4}$ Sol. (i) ${\left\{{\left(\frac{1}{3}\right)}^{-1}-{\left(\frac{1}{4}\right)}^{-1}\right\}}^{-1}={\left(3^1-4^1\right)}^{-1}=(-1{)}^{-1}$ $=\frac{1}{(-1{)}^1}=\frac{1}{-1}=-1$ (ii) ${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-4}$ $$\begin{gathered} ={\left(\frac{8}{5}\right)}^7\times {\left(\frac{8}{5}\right)}^{-4} \\ ={\left(\frac{8}{5}\right)}^{7+(-4)}={\left(\frac{8}{5}\right)}^{7-4}={\left(\frac{8}{5}\right)}^3=\frac{512}{125} \end{gathered}$$
7. Simplify (i) $\frac{25\times {\mathrm{t}}^{-4}}{5^{-3}\times 10\times {\mathrm{t}}^{-8}}(\mathrm{t}\ne 0)$ (ii) $\frac{3^{-5}\times 10^{-5}\times 125}{5^{-7}\times 6^{-5}}$ Sol. (i) $\frac{25\times {\mathrm{t}}^{-4}}{5^{-3}\times 10\times {\mathrm{t}}^{-8}}=\frac{25\times 5^3}{10}\times {\mathrm{t}}^{-4+8}$ $=\frac{5\times 5^3}{2}\times t^4=\frac{625}{2}{t}^4$ (ii) $\frac{3^{-5}\times 10^{-5}\times 125}{5^{-7}\times 6^{-5}}=\frac{3^{-5}\times 10^{-5}\times 5^3}{5^{-7}\times 6^{-5}}$ $=\frac{3^{-5}\times \left(2^{-5}\times 5^{-5}\right)\times 5^3}{5^{-7}\times \left(2^{-5}\times 3^{-5}\right)}$ $=5^{-5}\times 5^3\times 5^7$ $=5^{-5+3+7}=5^5=3125$

5.0Key Features and Benefits of Class 8 Maths Chapter 10 Exercise 10.1

• Introduction to Exponents: Exercise 10.1 helps students understand what exponents are and how they are used to represent large numbers in a shorter form.
• Simple Rules of Exponents: Students learn the basic laws of exponents like multiplication, division, and powers of powers in a clear and easy way.
• Makes Calculations Faster: Using exponents helps in doing quick calculations, especially with large numbers, making maths more fun and manageable.
• Builds a Strong Base for Higher Classes: Understanding these rules now makes it easier to handle more advanced topics in algebra and science later.
• Based on NCERT Syllabus: All questions and explanations follow the official NCERT textbook, ensuring students are well-prepared for exams.