Exponents represent repeated multiplication of a number by itself. The laws of exponents are mathematical rules that explain how to perform operations on expressions involving powers.
The important laws of exponents are given as:
Product Rule
In the Product of Powers Rule, if two numbers with the same bases and different exponents are multiplied, then the exponents of the base are added to find the product. It is represented as xm × xn = x(m+n)
Example: 52 × 53 = ?
Keep the base values the same because they're both five, and then add the exponents together (2+3).
52 × 53 = 52+3 = 55
To get the answer, multiply five by itself five times.
55 = 5 × 5 × 5 × 5 × 5 = 3125
Quotient Rule
In the Quotient of Powers Rule, if two numbers with the same bases and different exponents are divided, then the exponents of the base are subtracted to find the quotient. It is represented as xa÷xb = x(a-b)
Example: 45 ÷ 43 = ?
Solution:
45 ÷ 43 =?
Because both bases in this equation are four, they remain the same. Then subtract the divisor from the dividend using the exponents.
45 ÷ 43 = 45-3 = 42
Finally, if necessary, simplify the equation.
42 = 4 × 4 = 16
Power of a Power Rule
In the Power of a Power Rule, if a number raised to some power is again raised to some power, then the two powers will be multiplied. It is represented as (xm)n = xm×n
Example: (23)2 = ?
Solution:
Multiply the exponents
23×2
= 26
= 64
Power of a Product Rule
When two bases with the same exponent are multiplied, the bases can be multiplied first, and the common exponent is applied to the product.
It is represented as (xm × ym) = (xy)m
Example: 23×33 = ?
Solution:
Since the bases are different and the power is same then multiply the bases and raise it to the common power.
Therefore, 23×33 = (2×3)3 = 63 = 216
Example: (2×3)3 = ?
Solution:
In this case separate the same power to individual bases.
Hence, (2×3)3 = 23×33 = 8×27 = 216
Power of a Quotient Rule
The Power of a Quotient Rule states that when a quotient is raised to a power, the exponent is applied to both the numerator and denominator.
It is represented as:
Similarly:
Example: Simplify
Solution:
Using the Power of a Quotient Rule:
\frac{6^4}{3^4} = (\frac{6}{3})^4 = 24 = 16
Zero Power Rule
In the Zero Power Rule, if any base is raised to the power zero, then the result will be 1. This can be represented as x0 = 1.
Suppose we have to prove x0 = 1.
x0 = xn-n , where (0 = n-n)
From the Quotient of Power Rule, we know that if the base are same then we subtract the exponents while finding the quotient; the vice versa of Quotient of Power Rule also holds true.
⇒ xn-n = xn/xn = 1
Hence, x0 = 1.
Example: (1001)0 =?
As per Zero Power Rule, any number raised to power zero results the value 1.
(1001)0 = 1
Negative Exponent Rule
The Negative Exponent Rule states that when a number is raised to a negative exponent, we take the reciprocal of the base and change the exponent to a positive value.
It is represented as:
Similarly:
Example: (2/3)-2 =?
Solution:
Since, the exponent is negative the base is converted to its reciprocal.
⇒ (2/3)-2 = (3/2)2 = 32/22 = 9/4
Root Form Rule
The Root Form of Power Rule states that a power expressed as a fraction can be written in root form. The denominator of the fractional exponent represents the root, while the numerator represents the power.
It is represented as:
Here, a is the base of the exponent, and 1/n is the exponent in fractional form.
Example: Simplify (8)1/3
= (8)1/3 = ∛(8)
= ∛(2×2×2)
= 2
Other Rules of Exponents
Apart from the standard exponent rules, the following properties should also be kept in mind:
- If a negative number is raised to even number power then the result will be positive and if a negative number is raised to odd number power then the result is always negative. For example (-2)4 = 16 and (-2)5 = -32.
- If 1 is raised to any power then the result will be always 1. For example, 13 = 1, 11001 = 1.
- If any number except 1 is raised to power infinity then the result will be infinity. 2∞ = ∞