Laws of exponents: what they are and examples

What are the laws of exponents?

The laws of exponents are the set of rules established to solve mathematical operations with powers.

The power or empowerment consists of the multiplication of a number by itself several times, and they are represented graphically as follows: xy.

The number to be multiplied by itself is called the base and the number of times to multiply it is called the exponent, which is smaller and should be located to the right and above the base.

For example,

Now, in addition, subtraction, multiplication and division operations with one or more powers, how to proceed? The laws of exponents guide us to solve these operations in the simplest possible way. Let's see.

1) Zero power

1) Every number raised to 0 equals 1.

For example, x ⁰ = 1

5 ⁰ = 1

37 ⁰ = 1

2) Power at 1

Every number raised to 1 is equal to itself.

For example, x ¹ = x

30 ¹ = 30

45 ¹ = 45

3) Multiplication of powers with the same base

The product of powers with an identical base is equal to a power with the same base, raised to the sum of the exponents.

For example, 2 ⁴ · 2 ² · 2 ⁴ = 2 ^{(4 + 2 + 2)} = 2 ⁸

4) Power division with the same base

When powers with the same base and different exponents are divided, the quotient is equal to another power with the same base raised to the sum of the exponents.

For example, 4 ⁴: 4 ² = 4 ^{(4 - 2)} = 4 ²

5) Multiplication of powers with the same exponent

The product of two or more different powers with the same exponent is equal to the product of the bases raised to the same exponent.

For example:

3 ² · 2 ² · 3 ² = (3 · 2 · 3) ² = 18 ²

6) Division of powers with the same exponent

The quotient between two powers with different bases and the same exponent results in the quotient of the bases raised to the same exponent.

For example, 8 ²: 2 ² = (8: 2) ² = 4 ²

7) Power of a power

The power of one power results in another power with the same base raised to the product of the exponents.

For example:

(8 ³) ³ = 8 ^{(3 · 3)} = 8 ⁹

You may also be interested in Laws of exponents and radicals.