The laws of exponents are mathematical rules that explain how to perform operations on expressions involving powers.
These rules help simplify exponential expressions and make calculations involving multiplication, division, and powers more systematic and efficient.
Product of Powers 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 of Powers 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:
(23)2=?
Multiply the exponents together in equations like the one above while keeping the base constant.
23×2 = 26
However, we must remember that
(2^3)^2 ≠ 2^(3^2) , since(2^3)^2 = 2^6 while2^(3^2) = 2^9 , because in2^(3^2) only the exponent 3 is raised to 2 and not the entire base2^3 .
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
In the Power of a Quotient Rule, if two different bases with the same power are divided, then the result is the quotient of the bases raised to the same power. This is represented as xm/ym = (x/y)m. In this case, the vice versa is also true, i.e., if both the numerator and denominator are raised to the same power, then the power is distributed to both the numerator and denominator individually. It can be represented as (x/y)m = xm/ym.
Example: Simplify 64/34.
Solution:
In this case, find the quotient of the bases and raise common power to it.
64/34 = (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. The zero power rule can be understood from the following description
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
In the Negative Exponent Rule, if a number is raised to a negative exponent, then we convert the base to its reciprocal, and the power is changed to positive. The vice versa is also true, i.e., if the exponent is positive and if the base is converted to its reciprocal, then the exponent is changed to a negative value. It can be represented as (x/y)-m = (y/x)m
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
Fractional Exponent Rule
The fractional exponent rule is a rule that is used to solve fractional exponents or exponents that are in fractional form. An exponent in fractional form is written as a1/n and is read as the nth root of a. It is also represented as,
a1/n = n√(a)
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∞ = ∞
Laws of Exponents and Logarithms
The laws of exponents and logarithms are basic mathematical rules used to simplify expressions involving powers and logarithms, making calculations easier and more systematic.
The essential laws are summarized below.
Also Check
Solved Examples
Example 1: What is the simplification of 73 × 71?
Solution:
73 × 71 = 73+1 = 74
Example 2: Simplify and find the value of 102/52.
Solution:
We can write the given expression as;
102/52= (10/5)2 = 22 = 4
Example 3: Find the value of (256)3/4
Solution:
(256)3/4 = (44)3/4 = 44×(3/4) = 43 = 64
Example 4: Find the value of 7-3
Solution:
7-3 = (1/7)3 = 13/73 = 1/343
Example 5: Find the value of x if 125 = 25/5x
Solution:
We have 125 = 25/5x
⇒ 53 = 52/5x
⇒ 53 = 52-x
Now the quantity is the same on both sides and bases are also the same, hence, exponents will also be the same.
⇒ 3 = 2-x
⇒ x = 2-3 = -1
Practice Problems
1. (23)4
2. (53*52)/54
3. Solve for x in the equation 32x = 81
4. If 2a = 8 and 2b = 32, find the value of a + b.
5. Solve 2-3 × 4-1
6. Solve for x in the equation: 52x + 1 = 125
7. If 16x-1 = 8, find the value of x.
8. (27)2/3
9. Simplify (9/4)(3/2)
10. Solve for x in the equation: 22x-1 = 8x + 1