Study MaterialsCBSE NotesExponents 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)

    Exponents and Powers Class 7 Notes Maths Chapter 13 1
    (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.

    Exponents and Powers Class 7 Notes Maths Chapter 13 2

    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
    Exponents and Powers Class 7 Notes Maths Chapter 13 3
    Exponents and Powers Class 7 Notes Maths Chapter 13 4

    Numbers with Exponent Zero: If a be any non-zero integer, then, a0 = 1
    Exponents and Powers Class 7 Notes Maths Chapter 13 5

    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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