Laws of Exponents – Definition, Rules & Examples | Testbook
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In mathematics, there are several important rules called the laws of exponents. These rules help us solve problems that involve multiplying or dividing the same number many times. Using these laws makes solving such problems much easier and faster. In this article, we will look at the six main laws of exponents, along with clear and simple examples to help you understand how they work.
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What are Exponents?
An exponent tells us how many times to multiply a number by itself. For example, if we multiply 9 by itself three times, we write it as 9³. Here, the small number 3 is called the exponent, and it shows how many times we multiply 9. The number 9 is called the base because it’s the number being multiplied. Simply put, exponents (also called powers) tell us how many times to use the base number in multiplication. If the exponent is 2, it means the base number is multiplied by itself twice. Below are some examples to help understand this better:
- 64 = 6×6×6×6
- 115 = 11×11×11×11×11
- 173 = 17 × 17 × 17
If a number ‘b’ is multiplied by itself n-times, it is represented as bn, where b is the base and n is the exponent.
Rules of Exponents With Examples
As we have discussed, there are several laws or rules for exponents. The essential laws of exponents are as follows:
- am × an = am+n
- am / an = am-n
- (am)n = amn
- an / bn = (a/b)n
- a0 = 1
- a-m = 1/am
Now, let's discuss each of these laws in detail, complete with examples.
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Product With the Same Bases
When multiplying numbers with the same base, you keep the base the same and add their exponents. This rule makes it easier to simplify expressions involving repeated multiplication of the same number.
- am × an = am+n
where m and n are real numbers.
Example 1: Simplify 44 × 42
Solution: 44 × 42 = 44+2 = 46
Example 2: Simplify (−3)-2 × (−3)-4
Solution: (−3)-2 × (−3)-4 = (-3)-2-4 = (-3)-6
Quotient with Same Bases
When dividing numbers with the same base, you keep the base and subtract the exponent of the denominator from the exponent of the numerator. This helps simplify expressions where the same number is divided with different powers.
- am / an = am-n
Example 3: Simplify 5-3 / 5-1
Solution: 5-3 / 5-1 = 5-3+1 = 5-2 = 1/25
Power Raised to a Power
When a power is raised to another power, you multiply the exponents while keeping the same base. This rule helps simplify expressions with multiple layers of exponents.
- (am)n = amn
Example 4: Express 162 as a power with base 4.
Solution: We have, 4×4 = 16 = 42. Therefore, 162 = (42)2 = 44.
Product to a Power
When a product is raised to a power, you apply the exponent to each factor inside the parentheses. This means you raise every number in the product to that power.
- an bn = (ab)n
Example 5: Simplify and write the exponential form of: 1/6 x 3-2
Solution: We can write, 1/6 = 2-2. Therefore, 2-2 x 3-2 = (2 x 3)-2 = 6-2.
Quotient to a Power
When a quotient (a fraction) is raised to a power, you raise both the numerator and the denominator to that power separately. This makes it easier to simplify expressions involving powers of fractions.
- an / bn = (a/b)n
Here, a and b are non-zero numbers and n is an integer.
Example 6: Simplify the expression and find the value: 252 / 52
Solution: We can write the given expression as; (25/5)2 = 52 = 25
Zero Power
Any non-zero number raised to the power of zero equals one. This rule helps simplify expressions where the exponent is zero.
a0 = 1
Here, ‘a’ is any non-zero number.
Example 7: What is the value of 70 + 32 + 60 + 81 - 41?
Solution: 70 + 32 + 60 + 81 - 41 = 1 + 9 + 1 + 8 - 4 = 15
Negative Exponent Rule
A negative exponent means you take the reciprocal of the base and then raise it to the positive exponent. This rule helps simplify expressions with negative powers.
a-m = 1/am
Example 8: Find the value of 3-2.
Solution:
Here, the exponent is a negative value (i.e., -2). Thus, 3-2 can be written as 1/32, which equals 1/9.
Fractional Exponent Rule
A fractional exponent means taking a root of the base. The numerator is the power, and the denominator is the root.
Here, a is called the base, and 1/n is the exponent, which is in the fractional form. Thus, a1/n is said to be the nth root of a.
Example 9: Simplify: 811/2
Solution:
Here, the exponent is in fractional form. (i.e., ½). According to the fractional exponent rule, 811/2 can be written as √81, which equals 9.
Exponent Rules Chart
Name of Exponent Rule
Rule (Formula)
Zero Exponent Rule
a⁰ = 1
Identity Exponent Rule
a¹ = a
Product Rule
aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient Rule
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Negative Exponents Rule
a⁻ᵐ = 1 / aᵐ
(a/b)⁻ᵐ = (b/a)ᵐ
Power of a Power Rule
(aᵐ)ⁿ = aᵐⁿ
Power of a Product Rule
(ab)ᵐ = aᵐ × bᵐ
Power of a Quotient Rule
(a/b)ᵐ = aᵐ ÷ bᵐ
|
Name of Exponent Rule |
Rule (Formula) |
|
Zero Exponent Rule |
a⁰ = 1 |
|
Identity Exponent Rule |
a¹ = a |
|
Product Rule |
aᵐ × aⁿ = aᵐ⁺ⁿ |
|
Quotient Rule |
aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
|
Negative Exponents Rule |
a⁻ᵐ = 1 / aᵐ (a/b)⁻ᵐ = (b/a)ᵐ |
|
Power of a Power Rule |
(aᵐ)ⁿ = aᵐⁿ |
|
Power of a Product Rule |
(ab)ᵐ = aᵐ × bᵐ |
|
Power of a Quotient Rule |
(a/b)ᵐ = aᵐ ÷ bᵐ |
Practice Problems on Laws of Exponents
Simplify the following expressions using the laws of exponents:
- (32)3
- 32 × 35
- 4-3
- 491/2
- 80 × 23
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Frequently Asked Questions For Law of Exponents
What is an exponent?
An exponent tells us how many times a number (called the base) is multiplied by itself. Example: 23=2×2×2=8
What is the exponent rule for dividing the same base?
Subtract the powers:a^m ÷ a^n = a^{m−n}
What happens if the exponent is 1?
Any number raised to the power 1 is the number itself. Example: 7^1 = 7
Can the base be a variable?
Yes, exponents can be applied to variables as well as numbers. Example: x^3 × x^2 = x^{5}
Can we multiply two exponents with different bases but same power?
Yes,Example: a^n × b^n = (ab)^n
What does zero exponent mean?
Any non-zero number raised to the power of 0 is always 1. Example: 50=1
How are exponents used in real life?
Exponents are used in science (like measuring area, volume, population growth), finance (compound interest), and technology (data storage sizes, like MB, GB).