NCERT Solutions for Class 7 Maths Chapter 11: Exponents and Powers — Free PDF

Chapter Overview: Exponents and Powers

Exponents provide a concise way to write very large (or very small) numbers. Instead of writing $2 \times 2 \times 2 \times 2 \times 2$, you write $2^5$. This compact notation is not just a convenience — it is essential for science, engineering, and higher mathematics.

Distances in astronomy (the Sun is about $1.5 \times 10^{11}$ m from Earth), sizes of atoms (a hydrogen atom is about $5.3 \times 10^{-11}$ m), and even the national debt of countries are all expressed using powers of 10. This chapter teaches you the rules (laws of exponents) that make working with these numbers efficient.

The chapter has 3 exercises covering basics of exponents, laws of exponents, and standard form (scientific notation). This chapter is foundational for algebra in higher classes, where exponents are used with variables. Students who master the laws of exponents here will find algebraic expressions much easier in Class 8 and beyond.

Key Concepts and Definitions

Let us establish all the rules before solving problems.

Base, Exponent, and Special Cases

Exponent (Power): In $a^n$, the number $a$ is the base and $n$ is the exponent (or power or index). It means $a$ multiplied by itself $n$ times: $a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$.

Important special cases:
- $a^1 = a$ (any number to the power $1$ is itself)
- $a^0 = 1$ for any $a \neq 0$ (any non-zero number to the power $0$ is $1$)
- $1^n = 1$ for any $n$ ($1$ raised to any power is $1$)
- $(-1)^{\text{even}} = 1$ and $(-1)^{\text{odd}} = -1$

Laws of Exponents

For a non-zero base $a$ and integers $m$, $n$:

| Law | Formula | Example |
|---|---|---|
| Product of powers | $a^m \times a^n = a^{m+n}$ | $2^3 \times 2^4 = 2^7$ |
| Quotient of powers | $a^m \div a^n = a^{m-n}$ | $5^7 \div 5^3 = 5^4$ |
| Power of a power | $(a^m)^n = a^{mn}$ | $(3^2)^4 = 3^8$ |
| Product to a power | $a^m \times b^m = (ab)^m$ | $2^3 \times 5^3 = 10^3$ |
| Quotient to a power | $\dfrac{a^m}{b^m} = \left(\dfrac{a}{b}\right)^m$ | $\dfrac{4^3}{2^3} = 2^3$ |
| Zero exponent | $a^0 = 1$ | $7^0 = 1$ |

Key restriction: The first three laws require the same base. You cannot use $a^m \times a^n = a^{m+n}$ if the bases are different.

Standard Form (Scientific Notation)

Standard form: A number written as $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.

For large numbers: Move the decimal left and make $n$ positive.
$384000 = 3.84 \times 10^5$ (decimal moved 5 places left).

For small numbers: Move the decimal right and make $n$ negative.
$0.00045 = 4.5 \times 10^{-4}$ (decimal moved 4 places right).

Exercise 11.1 — Basics of Exponents (Solved)

Exercise 11.1 covers expressing numbers in exponential form, evaluating powers, and prime factorisation.

Q1-Q2: Exponential Form and Evaluation

Q1. Express in exponential form:
- $5 \times 5 \times 5 \times 5 = 5^4$
- $(-3) \times (-3) \times (-3) = (-3)^3 = -27$
- $t \times t \times t \times t \times t = t^5$

Q2. Evaluate:
- $2^6 = 64$
- $(-2)^4 = 16$ (even power of negative = positive)
- $(-2)^5 = -32$ (odd power of negative = negative)

Key rule: Even power of a negative number is positive; odd power is negative.

Q3-Q4: Expressing as Powers and Comparison

Q3. Express $729$ as a power of $3$.
$729 \div 3 = 243$, $243 \div 3 = 81$, $81 \div 3 = 27$, $27 \div 3 = 9$, $9 \div 3 = 3$, $3 \div 3 = 1$.
We divided by $3$ exactly $6$ times, so $729 = 3^6$.

Q4. Which is greater: $2^3$ or $3^2$?
$2^3 = 8$, $3^2 = 9$. So $3^2 > 2^3$.

Q5: Prime Factorisation in Exponential Form

Express each as a product of prime factors in exponential form:

- $108 = 2^2 \times 3^3$ (since $108 = 4 \times 27 = 2^2 \times 3^3$)
- $360 = 2^3 \times 3^2 \times 5$ (since $360 = 8 \times 9 \times 5$)
- $1600 = 2^6 \times 5^2$ (since $1600 = 64 \times 25$)

Method: Divide repeatedly by the smallest prime factor (2, then 3, then 5, etc.) and count the occurrences.

Exercise 11.2 — Laws of Exponents (Solved)

Exercise 11.2 applies the laws of exponents to simplify expressions.

Q1-Q3: Product, Quotient, and Power of a Power

Q1. Simplify $2^3 \times 2^5$.
Using $a^m \times a^n = a^{m+n}$: $2^3 \times 2^5 = 2^8 = 256$.

Q2. Simplify $\dfrac{5^7}{5^3}$.
Using $a^m \div a^n = a^{m-n}$: $5^{7-3} = 5^4 = 625$.

Q3. Simplify $(3^2)^4$.
Using $(a^m)^n = a^{mn}$: $3^{2 \times 4} = 3^8 = 6561$.

Q4-Q5: Zero Exponent and Combined Laws

Q4. Evaluate $7^0 + 3^0 + 5^0$.
$1 + 1 + 1 = 3$. Any non-zero number raised to power $0$ equals $1$.

Q5. Simplify $\dfrac{2^5 \times 3^5}{6^3}$.

Q6-Q8: Simplification and Finding x

Q6. Simplify $\dfrac{3^4 \times 3^2}{3^5}$.
$\dfrac{3^{4+2}}{3^5} = \dfrac{3^6}{3^5} = 3^1 = 3$.

Q7. Simplify $\dfrac{2^4 \times 5^2 \times 7}{8 \times 25}$.
Convert to prime bases: $8 = 2^3$, $25 = 5^2$.
$\dfrac{2^4 \times 5^2 \times 7}{2^3 \times 5^2} = 2^{4-3} \times 5^{2-2} \times 7 = 2 \times 1 \times 7 = 14$.

Q8. Find $x$: $2^x = 32$.
$32 = 2^5$, so $2^x = 2^5 \implies x = 5$.

Exercise 11.3 — Standard Form (Solved)

Exercise 11.3 covers expressing numbers in standard form and real-world applications.

Q1-Q3: Converting to Standard Form

Q1. Express $3{,}84{,}000$ in standard form.
$3{,}84{,}000 = 3.84 \times 10^5$ (moved decimal 5 places left).

Q2. Express $4.7 \times 10^4$ in usual form.
$4.7 \times 10^4 = 47{,}000$ (moved decimal 4 places right).

Q3. Express $5{,}00{,}00{,}000$ in standard form.
$5{,}00{,}00{,}000 = 5 \times 10^7$.

Q4: Speed of Light Problem

Problem: The speed of light is $3 \times 10^8$ m/s. The distance from the Sun to Earth is $1.5 \times 10^{11}$ m. How long does light take to travel from the Sun to Earth?

Solution:

This is about $8$ minutes and $20$ seconds.

Q5-Q6: More Standard Form Problems

Q5. Simplify and express in standard form: $25 \times 10^5 \times 4 \times 10^3$.
$25 \times 4 \times 10^{5+3} = 100 \times 10^8 = 1 \times 10^{10}$.

Note: $100 = 1 \times 10^2$, so $100 \times 10^8 = 10^{10}$.

Q6. Express $0.00045$ in standard form.
$0.00045 = 4.5 \times 10^{-4}$ (moved decimal 4 places right, so exponent is negative).

Worked Examples — Additional Practice

More challenging examples for exam preparation.

Example 1: Simplifying with Negative Base

Problem: Simplify $\dfrac{(-2)^5 \times (-2)^3}{(-2)^6}$.

Solution:

Example 2: Finding x with Fractional Base

Problem: Find $x$ if $\left(\dfrac{3}{4}\right)^x = \dfrac{27}{64}$.

Solution:
$\dfrac{27}{64} = \dfrac{3^3}{4^3} = \left(\dfrac{3}{4}\right)^3$

So $x = 3$.

Example 3: Verifying a Common Error

Problem: Is $2^3 \times 3^2 = 6^5$? Justify.

Solution:
$2^3 \times 3^2 = 8 \times 9 = 72$.
$6^5 = 7776$.
Clearly $72 \neq 7776$, so $2^3 \times 3^2 \neq 6^5$.

The law $a^m \times b^m = (ab)^m$ requires the same exponent. Here the exponents are different ($3$ and $2$), so this law does not apply.

Example 4: Earth vs Moon Mass

Problem: The mass of the Earth is approximately $5.97 \times 10^{24}$ kg and the mass of the Moon is approximately $7.35 \times 10^{22}$ kg. How many times heavier is the Earth?

Solution:

The Earth is approximately $81$ times heavier than the Moon.

Example 5: Complex Simplification

Problem: Simplify $\dfrac{2^5 \times 3^3 \times 5^2}{4^2 \times 9 \times 25}$.

Solution:
Convert to prime bases: $4 = 2^2$, $9 = 3^2$, $25 = 5^2$.

Common Mistakes to Avoid

Mistake 1: Adding exponents when bases are different.
$2^3 \times 3^2 \neq 6^5$. The law $a^m \times a^n = a^{m+n}$ works ONLY when the base is the same.

Mistake 2: Confusing $(-2)^4$ with $-2^4$.
$(-2)^4 = 16$ (the negative is inside the parentheses). $-2^4 = -(2^4) = -16$ (only $2$ is the base). Parentheses make a critical difference.

Mistake 3: Thinking $a^0 = 0$.
$a^0 = 1$ for any non-zero $a$, not $0$.

Mistake 4: Multiplying exponents instead of adding.
$2^3 \times 2^4 = 2^7$ (add), not $2^{12}$ (that would be $(2^3)^4$).

Mistake 5: Standard form errors.
The coefficient must be between $1$ and $10$. Writing $38.4 \times 10^4$ is NOT standard form. It should be $3.84 \times 10^5$.

Practice Questions with Answers

Test yourself with these problems.

Q1: Simplification

Question: Simplify $\dfrac{(-3)^4 \times (-3)^2}{(-3)^5}$.

Answer: $\dfrac{(-3)^{4+2}}{(-3)^5} = \dfrac{(-3)^6}{(-3)^5} = (-3)^1 = -3$.

Q2: Find x

Question: Find $x$ if $4^x = 256$.

Answer: $256 = 4^4$ (since $4 \times 4 \times 4 \times 4 = 256$). So $x = 4$.
Alternatively: $4^x = (2^2)^x = 2^{2x}$ and $256 = 2^8$, so $2x = 8$, $x = 4$.

Q3: Standard Form

Question: Express $0.0000067$ in standard form.

Answer: $0.0000067 = 6.7 \times 10^{-6}$ (decimal moved 6 places right).

Q4: Prime Factorisation

Question: Express $3600$ in exponential form using prime factors.

Answer: $3600 = 36 \times 100 = 6^2 \times 10^2 = (2 \times 3)^2 \times (2 \times 5)^2 = 2^4 \times 3^2 \times 5^2$.

Exam Tips for Exponents and Powers

Key Takeaways

- $a^n = a \times a \times \ldots \times a$ ($n$ times).
- Laws: $a^m \times a^n = a^{m+n}$, $a^m \div a^n = a^{m-n}$, $(a^m)^n = a^{mn}$.
- $a^0 = 1$ for any $a \neq 0$. The expression $0^0$ is undefined.
- $(-1)^{\text{even}} = 1$, $(-1)^{\text{odd}} = -1$.
- Standard form: $a \times 10^n$ where $1 \leq a < 10$.
- $a^m \times b^m = (ab)^m$ — this law requires the same exponent, not the same base.
- The first three laws (product, quotient, power of a power) require the same base.
- To find $x$ in $a^x = b$, express $b$ as a power of $a$ and compare exponents.

Ten Common Mistakes Students Make in Exponents and Powers

Exponents is a chapter where a few rules control almost every question. Mixing up those rules costs marks fast. Here are the ten mistakes you need to stop making.

Mistake 1: Adding exponents across different bases
❌ Wrong: $2^3 \times 3^2 = 6^5$.
✅ Correct: $2^3 \times 3^2 = 8 \times 9 = 72$. The product rule works only with the same base.

Mistake 2: Confusing $(-2)^4$ with $-2^4$
❌ Wrong: Both equal $-16$.
✅ Correct: $(-2)^4 = 16$. Without brackets, $-2^4 = -(2^4) = -16$.

Mistake 3: Thinking $a^0 = 0$
❌ Wrong: $5^0 = 0$.
✅ Correct: $5^0 = 1$. Any non zero number raised to the zero power is $1$.

Mistake 4: Multiplying exponents when you should add
❌ Wrong: $2^3 \times 2^4 = 2^{12}$.
✅ Correct: $2^3 \times 2^4 = 2^7$. You add exponents when multiplying same bases.

Mistake 5: Wrong standard form with coefficient bigger than $10$
❌ Wrong: $38.4 \times 10^4$.
✅ Correct: $3.84 \times 10^5$. The coefficient must satisfy $1 \le a < 10$.

Mistake 6: Forgetting to square or cube correctly
❌ Wrong: $2^5 = 10$.
✅ Correct: $2^5 = 32$. The exponent counts how many times the base is multiplied, not the result of multiplication.

Mistake 7: Applying the power of a power rule wrongly
❌ Wrong: $(2^3)^4 = 2^7$.
✅ Correct: $(2^3)^4 = 2^{12}$. You multiply the exponents when raising a power to a power.

Mistake 8: Confusing $a^m + a^n$ with $a^{m+n}$
❌ Wrong: $2^3 + 2^4 = 2^7$.
✅ Correct: $2^3 + 2^4 = 8 + 16 = 24$. The product rule does not apply to addition.

Mistake 9: Ignoring the sign of the exponent when moving a decimal
❌ Wrong: $0.00034 = 3.4 \times 10^4$.
✅ Correct: $0.00034 = 3.4 \times 10^{-4}$. Small numbers have negative exponents in standard form.

Mistake 10: Evaluating $-1$ to an even power as $-1$
❌ Wrong: $(-1)^8 = -1$.
✅ Correct: $(-1)^8 = 1$. Even powers of $-1$ equal $1$; odd powers equal $-1$.

Previous Year Questions (PYQs) with Solutions

These are CBSE Class 7 style Exponents and Powers questions from the past few years.

Q1. Evaluate $2^5 \times 2^3$. `(CBSE 2021, 2 marks)`
Solution: $2^{5+3} = 2^8 = 256$.
Answer: $256$

Q2. Simplify $\tfrac{3^7}{3^4}$. `(CBSE 2022, 2 marks)`
Solution: $3^{7-4} = 3^3 = 27$.
Answer: $27$

Q3. Express $0.000056$ in standard form. `(CBSE 2023, 2 marks)`
Solution: Move the decimal $5$ places to the right. $0.000056 = 5.6 \times 10^{-5}$.
Answer: $5.6 \times 10^{-5}$

Q4. Find $x$ if $5^x = 625$. `(CBSE 2022, 3 marks)`
Solution: $625 = 5^4$, so $x = 4$.
Answer: $4$

Q5. Simplify $\tfrac{(2^5)^2 \times 2^3}{2^9}$. `(CBSE 2023, 3 marks)`
Solution:
$(2^5)^2 = 2^{10}$.
$2^{10} \times 2^3 = 2^{13}$.
$\tfrac{2^{13}}{2^9} = 2^4 = 16$.
Answer: $16$

Q6. Express the number $3{,}40{,}000$ in standard form. `(CBSE 2021, 2 marks)`
Solution: $3.4 \times 10^5$.
Answer: $3.4 \times 10^5$

Q7. Express $2^5 \times 3^4$ in usual form. `(CBSE 2024, 3 marks)`
Solution: $32 \times 81 = 2592$.
Answer: $2592$

Q8 (HOTS). If $2^x \times 2^4 = 2^{10}$, find the value of $x$. Then find the value of $(2^x)^2$. `(CBSE 2024, 4 marks)`
Solution:
$2^{x + 4} = 2^{10}$, so $x + 4 = 10$, $x = 6$.
$(2^6)^2 = 2^{12} = 4096$.
Answer: $x = 6$, $(2^x)^2 = 4096$.

Practice This Chapter on SparkEd

Exponents and Powers is the chapter where you either learn the rules cold or spend the rest of Class 8 and 9 struggling. The SparkEd Exponents and Powers module is built to make the rules automatic.

Inside you will find more than 60 practice questions across three levels. Level 1 drills the basics: evaluating small powers, identifying the base and exponent, and applying each law in isolation. Level 2 blends the laws and pushes you to simplify expressions that require two or three rules in sequence. Level 3 is the CBSE exam zone with standard form problems, find the value of $x$ puzzles, and HOTS questions that involve prime factorisation and comparing large numbers.

The instant AI feedback catches the most common errors in this chapter, like adding exponents across different bases or forgetting that $a^0 = 1$, and shows the exact correction. The gamified progress system keeps you engaged and the adaptive engine serves you the kinds of problems you struggle with the most.

Practice Exponents and Powers on SparkEd →

Download Exponents and Powers worksheet

Twenty minutes a day for a week and the laws will feel like second nature.