Exponents and Powers Class 7 Notes Maths Chapter 13

CBSE Class 7 Maths Notes Chapter 13 Exponents and Powers

Exponents
We can write large numbers in a short form using exponents.
For example: 10,000 = 10 × 10 × 10 × 10 = 10⁴
Here, ‘10’ is called the base and ‘4’ the exponent. The number 10⁴ is read as 10 raised to the power of 4 or simply as the fourth power of 10.
10⁴ is called the exponential form of 10,000.

(1)^{any natural number} = 1

(-1)^{an odd natural number} = -1

(-1)^{an even natural number} = +1

a^m × aⁿ = a^m+n, where m and n are whole numbers and a (≠ 0) is an integer.
This formula can be used to write answers to above questions.

For any non-zero integer a,
a^m ÷ aⁿ = a^m-n where m and n are whole numbers and m > n.

For any non-zero integer a,
(a^m)ⁿ = a^mn (where m and n are whole numbers)

For any non-zero integer a
a^m × b^m = (ab)^m (where m is any whole number)

(where m is a whole number; a and b are any non-zero integers)

a⁰ = 1 (for any non-zero integer a)
Any number (except 0) raised to the power (or exponent) 0 is 1.

Decimal Number System
10,000 = 10⁴
1000 = 10³
100 = 10²
10 = 10¹
1 = 10⁰
We can write the expansion of a number using powers of 10 in the exponent form.

Expressing Large Numbers in the Standard Form
Large numbers can be expressed conveniently using exponents. Such a number is said to be in standard form if it can be expressed as k × 10^m, where 1 ≤, k < 10 and m is a natural number.

Note that, one less than the digit count (number of digits) to the left of the decimal point in a given number, is the exponent of 10 in the standard form.

For any rational number a and positive integer n, we define aⁿ as a × a × a × …… × a (n times). aⁿ is known as the nth power of a and is read as ‘a raised to the power n’. The rational a is called the base and n is called the exponent or power.
e.g. 10,000 = 10 × 10 × 10 × 10 = 10⁴.
10 is the base and 4 is the exponent.

Multiplying Powers with the Same Base: If a is any non-zero integer and whole numbers are m and n, then a^m × aⁿ = a^m+n
e.g. 2⁴ × 2²
a = 2, m = 4, n = 2
2⁴ × 2² = 2⁴⁺² = 2⁶

Dividing Powers with the Same Base: If a is any non-zero integer and m, n are the whole number, then a^m ÷ aⁿ = a^m-n
e.g. 2⁴ ÷ 2²
a = 2, m = 4, n = 2
2⁴ ÷ 2² = 2^4-2 = 2²

Taking Power of a Power: If a is any non-zero integer and m, n are whole numbers, (a^m)ⁿ = a^mn
e.g. (6²)⁴
a = 6, m = 2, n = 4
(6²)⁴ = (6)^2×4 = 6⁸.

Multiplying Powers with the Same Exponents: If a, b are two non-zero integers and m is any whole number, then
a^m × bⁿ = (a × b)^m
e.g. 2³ × 3³
a = 2, b = 3, m = 3
2³ × 3³ = (2 × 3)³ = 6³.

Dividing Powers with the Same Exponents: If a, b are two non-zero integers and m is a whole number, then

Numbers with Exponent Zero: If a be any non-zero integer, then, a⁰ = 1

Numbers with Negative Exponent: If a is any non-zero integer, then a^-1 = \(\frac { 1 }{ a }\)
e.g. 2^-5 = \(\frac { 1 }{ { 2 }^{ 5 } }\)

In decimal number system, the exponents of 10 start from a maximum value and go on decreasing from the left to right upto 0.
e.g. 45672 = 4 × 10000 + 5 × 1000 + 6 × 100 + 7 × 10 + 2 × 1
= 4 × 10⁴ + 5 × 10³ + 6 × 10² + 7 × 10¹ + 2 × 10⁰
It is called expanded form of a number.

Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form.
e.g. 56782 = 5.6782 × 10000 = 5.6782 × 10⁴.
It is the standard form of 56782.

Class 7 Maths Notes