Laws of Exponents: Definition, Exponents Rules & Examples

In mathematics, exponents are the powers widely used in solving algebraic problems. Laws of exponents states that the base is the variable that is repeatedly multiplied by itself. Exponents show the repeated number of times where the number can be multiplied. For example, 4 x 4 x 4, can be represented as 4³ where 3 is the exponent and 4 is the base. There are different types of rules or laws defined for the exponents.

Laws of exponents are used to simplify the problems in the field of science and mathematics. Students learn the laws of exponents in their secondary school in order to solve mathematical problems quickly. You can check NCERT Solutions for Class 7 Maths Chapter 13 for a better understanding of the concepts. We have provided detailed information on the laws of exponents in this article. Read on to find out about the definition, Exponent Rules with Examples.

What Are the Laws of Exponents?

In the 9th century, a Persian Mathematician Muhammad Musa introduced the square of numbers. Later, the term exponent was termed in 1544. Exponents are the powers that are used to simplify the multiplication and division of repeated numbers. Laws of exponents comprise two parts i.e., base and exponent. Exponents are used to representing the repeated multiplication of numbers by themselves.

For example. 6 x 6 x 6 x 6 x 6 = 6 ⁵ . Here, 6 is the base and 5 is the exponent.

Exponent Rules with Examples

Exponents have different rules set to simplify the process of multiplication and division of expressions. Therefore, the important laws of exponents are mentioned below:

a^m×aⁿ = a^m+n : This law of exponent is applicable if the product has the same bases. For example, 2⁵ × 2¹ = 2⁵⁺¹ = 2⁶

a^m/aⁿ = a^m-n: This law of exponent is applicable if the quotient has the same bases. For example, 8^-5/8^-3 = 8^-5-(-)3

(a^m)ⁿ = a^mn : This law is applicable if the power raised to power . For example, (3³)³ = 3⁹

aⁿ/bⁿ = (a/b)ⁿ : This law is applicable if the product is same with different powers. For example, 15³/5³ = (15/5)³= 3³ = 27

a⁰ = 1: This law is applicable if the power is zero, making its value equal to 1. For example, 3⁰ + 4² + 8⁰ + 2¹ – 9¹

a^-m = 1/a^m: This law is applicable if the power has negative exponent. For example, 2^-2 = 1/2²

Examples of Laws of Exponents

Some of the examples of exponents are given below:

Example 1: Express the following in exponential form: (A) 6 × 6 × 6 × 6 (B) t × t (C) b × b × b × b (D) 5 × 5 × 7 × 7 × 7 (E) 2 × 2 × a × a

Solution:

(A) 6 × 6 × 6 × 6 = 6 ⁴

(B) t × t = t ²

(D) 5 × 5 x × 7 × 7 × 7 = 5 ² × 7 ³

(E) 2 × 2 × ?? × ?? = 2 ² × ? ²

Download NCERT Solutions for Class 7 Maths Chapter 13

Example 2: Using laws of exponents, simplify and write the answer in exponential form: (A) 3 ⁴ × 3 ⁴ × 3 ⁸ (B) 6 ¹⁵ ÷ 6 ⁴⁰ (C) a ³ × a ² (D) a ⁴ × b ⁴ (E) (3 ⁴ ) ³

Solution: (A) 3 ² × 3 ⁴ × 3 ⁸ = 3( ^{2 + 4+8} ) = 3 ¹⁴ (∵ a ^m × a ⁿ = a ( ^{m +n} ))

(B) 6 ¹⁵ ÷ 6 ¹⁰ = 6( ^15-10 ) = 6 ⁵ (∵ a ^m ÷ a ⁿ = a ( ^{m -n} ) )

(D) a ⁴ × b ⁴ = (a × b) ⁴ (∵ a ^m × b ^m = (a x b ^m)

(E) (3 ⁴ ) ³ = 3( ^{4 X3} ) = 3 ¹² (∵ (a ^m ) ⁿ = a ^{m xn}

FAQs

The frequently asked questions on laws of exponents are given below:

Q. What are exponents?
A. Exponents are the powers that are used to simplify the multiplication and division of repeated numbers.

Q. What are the parts of laws of exponent?
A. Laws of exponents comprises of two parts i.e., base and exponent. For example, 6 x 6 x 6 x 6 x 6 = 6 ⁵ . Here, 6 is the base and 5 is the exponent.

Q. What are rules of exponents?
A. The rules of exponents are as follows:
1. a^m×aⁿ = a^m+n
2. a^m/aⁿ = a^m-n
3. (a^m)ⁿ = a^mn
4. aⁿ/bⁿ = (a/b)ⁿ
5. a⁰ = 1
6. a^-m = 1/a^m

Q. What is the example of rule a^m×aⁿ = a^m+n ?
A. This law of exponent is applicable if the product has the same bases. For example, 2⁵ × 2¹ = 2⁵⁺¹ = 2⁶.

Now you are provided all the necessary information laws of exponents in this article. Your performance in the foundation exams will play a role in determining your admission into higher classes all the way to a good college! So, make the best use of NCERT CBSE solutions provided here and get a strong command over the NCERT syllabus.

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