Class 7 — Mathematics (NCERT)

Exponents and Powers

Class 7

✓By the end of this lesson students will be able to understand the concept of exponents and powers.
✓By the end of this lesson students will be able to apply the laws of exponents to simplify expressions.
✓By the end of this lesson students will be able to express large numbers in standard form.
✓By the end of this lesson students will be able to compare numbers expressed in exponential or standard form.

Key concepts

When a number is multiplied by itself repeatedly, we can write it in a shorter form called exponential form. For example, 2 × 2 × 2 × 2 × 2 can be written as 2^5. Here, '2' is called the base and '5' is called the exponent or power. The entire expression 2^5 is read as '2 raised to the power 5' or '2 to the power 5'. The exponent tells us how many times the base is multiplied by itself.

Laws of Exponents

These are rules that help us simplify expressions involving exponents. For any non-zero integers 'a' and 'b', and whole numbers 'm' and 'n':

Law 1: Multiplying Powers with the Same Base

When multiplying powers with the same base, we add the exponents.

a^m × a^n = a^(m+n)

Law 2: Dividing Powers with the Same Base

When dividing powers with the same base, we subtract the exponents (provided the exponent of the numerator is greater than or equal to the exponent of the denominator).

a^m ÷ a^n = a^(m-n) (where m ≥ n)

Law 3: Power of a Power

When a power is raised to another power, we multiply the exponents.

(a^m)^n = a^(m×n)

Law 4: Multiplying Powers with the Same Exponents

When multiplying powers with different bases but the same exponent, we multiply the bases and keep the exponent common.

a^m × b^m = (ab)^m

Law 5: Dividing Powers with the Same Exponents

When dividing powers with different bases but the same exponent, we divide the bases and keep the exponent common.

a^m ÷ b^m = (a/b)^m

Law 6: Zero Exponent

Any non-zero number raised to the power of zero is always 1.

a^0 = 1 (where a ≠ 0)

Standard Form (Scientific Notation)

Standard form is a way of writing very large or very small numbers using powers of 10. A number is said to be in standard form if it is expressed as k × 10^n, where 'k' is a decimal number such that 1 ≤ k < 10, and 'n' is a whole number. This makes it easier to read, understand, and compare large numbers.

Key facts to remember

1An exponent indicates how many times the base is multiplied by itself.
2a^m × a^n = a^(m+n)
3a^m ÷ a^n = a^(m-n) (where m ≥ n)
4(a^m)^n = a^(m×n)
5a^m × b^m = (ab)^m
6a^m ÷ b^m = (a/b)^m
7a^0 = 1 (for any non-zero base 'a')
8Standard form expresses a number as k × 10^n, where 1 ≤ k < 10 and 'n' is a whole number.

Worked examples

Example 1

Simplify: (2^3 × 3^4 × 4) / (3 × 32)

IFirst, express all numbers in terms of their prime factors:

II4 = 2^2

III32 = 2^5

IVSubstitute these into the expression:

V(2^3 × 3^4 × 2^2) / (3^1 × 2^5)

VIGroup terms with the same base in the numerator:

VII(2^(3+2) × 3^4) / (3^1 × 2^5)

VIII(2^5 × 3^4) / (2^5 × 3^1)

9Now, apply the division law of exponents (a^m / a^n = a^(m-n)):

102^(5-5) × 3^(4-1)

112^0 × 3^3

12Apply the zero exponent law (a^0 = 1):

131 × 3^3

14Calculate 3^3:

151 × (3 × 3 × 3)

161 × 27

Answer

Always express composite numbers as products of their prime factors before applying the laws of exponents.

Example 2

Express the number 3,84,00,000 in standard form.

ITo write a number in standard form, we need to place the decimal point such that there is only one non-zero digit to its left. For 3,84,00,000, the decimal point is implicitly at the end.

IIMove the decimal point to the left until it is after the first non-zero digit (3):

III3.8400000

IVCount the number of places the decimal point was moved. It was moved 7 places to the left.

VThe power of 10 will be 7 (positive, because the original number was large).

VISo, the number in standard form is 3.84 × 10^7.

Answer

3.84 × 10^7

For large numbers, the exponent of 10 is positive. For numbers between 0 and 1, the exponent would be negative (though negative exponents are typically covered in Class 8).

Example 3

Simplify: (5^2)^3 × 5^4 ÷ 5^7

IFirst, apply the 'power of a power' law: (a^m)^n = a^(m×n)

II(5^2)^3 = 5^(2×3) = 5^6

IIIThe expression becomes: 5^6 × 5^4 ÷ 5^7

IVNext, apply the 'multiplying powers with the same base' law: a^m × a^n = a^(m+n)

V5^6 × 5^4 = 5^(6+4) = 5^10

VIThe expression becomes: 5^10 ÷ 5^7

VIIFinally, apply the 'dividing powers with the same base' law: a^m ÷ a^n = a^(m-n)

VIII5^10 ÷ 5^7 = 5^(10-7)

95^3

10Calculate the value of 5^3:

115 × 5 × 5 = 25 × 5 = 125

Answer

125

Follow the order of operations (BODMAS/PEMDAS) when simplifying expressions involving multiple laws.

Common mistakes

✗Confusing the base and the exponent (e.g., thinking 2^3 is 2 × 3 instead of 2 × 2 × 2).
✗Incorrectly applying laws when bases are different (e.g., 2^3 × 3^2 ≠ 6^5).
✗Incorrectly applying laws when exponents are different (e.g., 2^3 × 2^4 = 2^12 instead of 2^7).
✗Forgetting that any non-zero number raised to the power of zero is 1 (e.g., writing 5^0 = 0).
✗Making errors in counting decimal places when converting to standard form, especially for large numbers.

Exam tips

★Memorise all the laws of exponents thoroughly. Write them down before starting to solve problems if allowed.
★Always express composite numbers as their prime factors before applying the laws of exponents to simplify expressions.
★Show all steps clearly in your working. This helps in identifying errors and earns partial marks even if the final answer is incorrect.
★Practice converting numbers to and from standard form regularly to avoid errors in decimal placement and powers of 10.

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