NCERT Solutions for Class 7 Maths Chapter 13 Exercise 13.1

NCERT Solutions for Class 7 Maths Chapter 13 Exercise 13.1 (Ex. 13.1) Exponents and Powers updated for academic session 2021-2022. All the solutions are given in PDF file format as well as videos format free to use without any format login.

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Class 7 Maths Chapter 13 Exercise 13.1 Solution

Class: 7Mathematics
Chapter: 13Exponents and Powers
Exercise: 13.1PDF and Videos Solution

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Exponents and Powers

An expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

Example:
5 x 5 x 5 x 5 as 5 and read it as 5 raised to the power 4. In 54, we call 5 the base and 4 the exponent (or index). The number of times a number is multiplied by itself is called the exponent.
Similarly, (-6) x (-6) x (-6) is written as (-6). In (-6)3, the base is (-6) and the exponent is 3.
This notation is called exponential form or power notation.
Similarly, the product of a rational number multiplied several times by itself can be expressed in the same notation.
Thus, (2/3)⁴ = 2/3 X 2/3 X 2/3 X 2/3
If a/b is a rational number, then
(a/b)² = 2/3 X 2/3 = (2 X 2)/(3 X 3)
In general, we have:
(a/b)ⁿ = (a)ⁿ /(b)ⁿ for every positive integer n

Express the following numbers as a product of powers of prime factors: (i) 81, (ii) 1000

(i) 81 = 3 x 3 x 3 x 3 = 3⁴
Thus, 81 = 3⁴
(ii) 1000 = 2 x 2 x 2 x 5 x 5 x 5 = 2³ x 5³
Thus, 1000 = 2³ x 5³

Reciprocal of a Rational Number:

If a/b is a nonzero rational number, then its reciprocal is
b/a and we write, (a/b)⁻¹ = b/a
Now, (a/b)ᵐ = aᵐ / bᵐ
Reciprocal of (a/b)ᵐ = (b/a)ᵐ

Write the reciprocal of each of the following in exponential form: (i) (2/5)⁶ (ii) (-3/7)⁸⁹ (iii) 3⁸

We know that the reciprocal of (a/b)ᵐ is (b/a)ᵐ
Therefore,
(i) Reciprocal of (2/5)⁶ = (5/2)⁶
(ii) reciprocal of (-3/7)⁸⁹ = (-7/3)⁸⁹
(iii) reciprocal of 3⁸ = (1/3)⁸

Laws of Exponents:

The laws of exponents on integers can be extended to rational numbers.
Thus, for any rational number a/b and positive integers m and n, we have:
(i) (a/b)ᵐ X (a/b)ⁿ = (a/b)ᵐ⁺ ⁿ
(ii) If m > n, then (a/b)ᵐ / (a/b) ⁿ = (a/b)ᵐ⁻n
(iii) If m < n, then (a/b)ᵐ / (a/b)ⁿ = 1/(a/b)ᵐ ⁻ ⁿ (iv) {(a/b) ᵐ }ⁿ = (a/b) ᵐⁿ (v) (a/b)⁻ ⁿ = 1/(a/b)ⁿ = 1/ (aⁿ /bⁿ) = bⁿ /aⁿ = (b/a) ⁿ Thus, (a/b)⁻ ⁿ = (b/a) ⁿ (vi) (a/b)₀ = 1 (vii) (a/b)⁻¹ = (b/a)

How are exponents used in daily life?

Another example of using exponents in real life is when you calculate the area of any square. If you say “My room is twelve foot by twelve-foot square”, you’re meaning your room is 12 feet × 12 feet — 12 feet multiplied by itself — which can be written as (12 ft)2. And that simplifies to 144 square feet.

How does the exponents relate to one another?

Adding the exponents is just a short cut! The “power rule” tells us that to raise a power to a power, just multiply the exponents. Here you see that 52 raised to the 3rd power is equal to 56. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents.

Simplify and express the result in exponential form: (i) (3/4) ⁷ x (3/4) ⁵ (ii) (-2/5) ¹¹ x (-2/5) ⁴

We have:
(i) (3/4) ⁷ x (3/4) ⁵ = (3/4) ⁷⁺⁵ = (3/4)¹²
(ii) (-2/5) ¹¹ x (-2/5) ⁴ = (ii) (-2/5) ¹¹⁺⁴ = (-2/5) ¹⁵

NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers Exercise 13.1
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