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Exponents

by Ron Kurtus (10 January 2008)

When you multiply a number or variable times itself numerous times, you can designate the result in exponential notation to make it more convenient to write. It is a form of algebraic shorthand. The number you are multiplying by itself is called the base and the number of times you multiply it by itself is called the exponent. You simply put the exponent as a small number at the upper right side of your base number. For example, if you would multiply 7 times itself 4 times, you would get 7 × 7 × 7 × 7 = 7⁴, where 7 is the base and 4 is the exponent. This is often called raising to the power.

Questions you may have are:

• What is the exponential notation for numbers?
• How are exponents designated for variables?
• What does raising to a power mean?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Exponential notation for numbers

If you multiply 5 times itself 6 times, or 5 × 5 × 5 × 5 × 5 × 5, you can write it as 5⁶, where 5 is the base number and 6 is the exponent. Certainly, 5⁶ is more convenient to write than 15,625.

You can also use exponents with decimal numbers. For example 1.3 × 1.3 = 1.3² and 0.2 × 0.2 × 0.2 = 0.2³.

Exponential notation for variables

You can use the exponential notation with variables. If x is a variable that represents a number, group of numbers or other variables, then x*x*x*x can be written as x⁴. In this case, x is the base and 4 is the exponent.

Note that we used * to denote multiplication instead of ×, since that multiplication sign can be confused with the letter x. Many algebra books use · as multiplication, but that also can be confused with the decimal point, such as 3·5 versus 3.5. The * sign is more common on web pages for multiplication.

It is possible to have a variable as both the base and the exponent, such as x^y. Thus, if we later say that x = 2 and y = 3, we would have x^y = 2³ = 8.

Note that sometimes on the Internet, you will see the ^ sign to indicate raised to a power. For example, b^5 = b⁵ and 10^3 = 10³.

Raised to the power

A number or variable is called raised to the power... when it is in the exponential form. For example, x⁴ is x raised to the 4th power and x¹⁰ is x raised to the 10th power. The same is try for numbers, where 5⁶ is 5 raised to the 6th power.

This way of describing exponential terms is not used when the exponent is 2 or 3. In such a case, the item is either a squared or cubed. For example, x² is usually called x-squared, since the area of a square is x*x. Following that logic, x³ is usually called x-cubed, since the volume of a cube is x*x*x.

Likewise, 5² is 5-squared and 5³ is 5-cubed.

Summary

When you multiply a number or variable times itself numerous times, you can designate the result in exponential notation to make it more convenient to write. You simply put the number of multiplications as a small number at the upper right side of your base number. It is often called raising to the power.

Answers to Readers' Questions

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Mini-quiz to check your understanding

1. What is a*a*a*a in exponential notation?

2. If a = 2 and b = 3, how much it a^b?

3. If x = 3, how much is x-cubed?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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