Exponents and Powers: Maths Class 7 Chapter-12

Key Features of NCERT Material for Class 7 Maths Chapter 12 – Algebraic Expressions

Quick revision notes

In Chapter 12 of Class 7 NCERT book: you must have learnt about Algebraic Expressions  In chapter 13:  you will learn about various forms of exponents and powers.

Introduction to Exponents and Powers

When the numbers given are very large like 54,32,00,00,000 then it is not easy to read them so we write them in the form of exponents.

Exponents make these numbers easy to read, write, understand and compare.

Exponents

To write the large numbers in short form, we use exponents.

The Expanded form of Natural Numbers

When we write the expanded form of a natural number then it can be written in exponential form.

Example

247983 = 2 × 100000 + 4 × 10000 + 7 × 1000 + 9 × 100 + 8 × 10 + 3 × 1

= 2 × 10⁵ + 4 × 10⁴ + 7 × 10³ + 9 × 10² + 8 × 10¹ + 3 × 1

Laws of Exponents

1. How to multiply powers with the same base?

If we have to multiply the powers which have the same base then we have to add the exponents.

a^m × aⁿ = a^{m + n}

2. How to divide powers with the same base?

If we have to divide the powers which have the same base then we have to subtract the exponents.

a^m/aⁿ = a^m-n   , m>n

3. How to take the power of a power?

If we have to take the power of a power then we have to multiply the exponents.

(a^m)ⁿ = a^m*n

4. How to multiply the powers with the same exponents?

If we have to multiply the powers where the base is different but exponents are same then we will multiply the base.

a^mb^m = (ab)^m

5. How to divide the powers with the same exponents?

If we have to divide the powers where the base is different but exponents are same then we will divide the base.

a^m/b^m = (a/b)^m

6. Numbers with Exponent Zero

Any number with zero exponents is equal to one irrespective of the base.

a^° = 1

7. Numbers with Exponent One

Any number with one as the exponent is equal to the number itself.

a¹ = a

8. Power with a Negative Exponent

Negative exponents can be converted into positive exponents.

1/aⁿ  = a^-n

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

am × an = am+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,

am ÷ an = am-n where m and n are whole numbers and m > n.

For any non-zero integer a,

(am)n = amn (where m and n are whole numbers)

For any non-zero integer a

am × bm = (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 × 10m, 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 am × an = am+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 am ÷ an = am-n

e.g. 24 ÷ 22

a = 2, m = 4, n = 2

24 ÷ 22 = 24-2 = 22

Taking Power of a Power: If a is any non-zero integer and m, n are whole numbers, (am)n = amn

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

am × 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 =

e.g. 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.