Exponents and Powers Class 7 Notes Maths Chapter 13

# 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

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(-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 = $$\frac { 1 }{ a }$$
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.

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