Powers and Exponents

An exponent is a number that indicates how many times a certain number, say b, is multiplied by itself. The expression 2 × 2 × 2 × 2 can be written as 2⁴, where 4 is the exponent and 2 is called the base. An exponent, which is also called a power, is written as a superscript of the base number. A base b raised to a power n is written as b ⁿ.

Exponent is a simple but powerful idea and can be used to create shortcuts in problems. To multiply 2⁴ by 2⁶, for instance, simply add the powers to find the product 2^{4 + 6}, or 2¹⁰.

What if 2⁴ is multiplied by 3⁶? The bases, 2 and 3, are different. In this case, the product cannot be found by adding the powers. The following are the three basic rules of exponents. Using these three laws, more properties of exponents can be found.

1. When multiplying two numbers with the same base, add the exponents: b ⁿ × b ^m = b ^{n + m}
   Example: 3³ × 3⁸ = 3^{3 + 8} = 3¹¹
2. When dividing two numbers with the same base, subtract the exponents:
   Example:
3. When raising a power to a power, multiply the exponents: (b ⁿ)^m = b ^{n × m}
   Example: (4³)² = 4^{3 × 2} = 4⁶.

Zero Exponent

An interesting rule involving exponents is that a number raised to zero power, say b ⁰, is equal to 1. This surprising result follows directly from Rule 2. Recall, a number divided by itself is 1.

Apply Rule 2.

Therefore, b ⁰ = 1. This means that any number raised to 0 is 1. Hence, 5⁰, 2⁰, and 31⁰ are all equal to 1.

Negative Exponent

What does a negative exponent mean? Here is another rule that also follows from Rule 2.

Using b ⁰ = 1 and b ¹ = b, can be expressed as follows.

Apply Rule 2 to the right-hand side.

Therefore,

So,

Fractional Exponent

A base raised to a fractional power, say b ^1/2, is another way to express the square root of b.

Therefore, . Similarly, b ^1/3 is another way to express the cube root of b.

Therefore, . In a general case, the n th root of b is b ^1/n.

Combining Rule 1 and the fractional exponent rule results in the following exponent property.

For instance, . The number within the parenthesis, square root of 4, is 2.

4^5/2 = (2)⁵

4^5/2 = 2 × 2 × 2 × 2 × 2

4^5/2 = 32

see also Radical Sign.

Rafiq Ladhani

Bibliography

Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.