Exponents

by Ron Kurtus (updated 18 January 2022)

Exponents are used in Algebra as a convenient way to designate multiplying a number or variable by itself numerous times.

The entity you are multiplying by itself is called the base and the number of times you multiply it by itself is called the exponent. To designate the expression, 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 by 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 include:

• 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.

Exponential notation for numbers

If you multiply 5 by 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. It is also called 5 raised to the 6th power.

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 commonly used on web pages for multiplication.

It is possible to have a variable as both the base and the exponent, such as xz. Thus, if we later assign values to the variables, such as x = 2 and z = 3, we would have xz = 23 = 8.

Raised to the power

When you put a number or variable in the exponential form, the item is often called raised to the power. For example, x4 can be called x raised to the 4th power. Likewise, x10 is x raised to the 10th power. The same is true for numbers, where 56 is 5 raised to the 6th power.

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.

Describing when the exponent is 2 or 3

When the exponent is 2, you would normally say that the base is squared. When the exponent is 3, you would say that the base is cubed. This comes from designating area or volume.

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 by 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.

Make things convenient

Resources and references

Ron Kurtus' Credentials

Websites

Exponents: Basic Rules - PurpleMath.com

Exponent Rules - RapidTables.com

Laws of Exponents - MathisFun.com

Exponents Calculator - CalculatorSoup.com

Algebra Resources

Books

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