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Explanation of Exponents by Ron Kurtus - Succeed in Understanding Algebra. Also refer to repeated multiplication, exponential, variable, algebraic, base, raising to a power, mathematics, math, maths, arithmetic, School for Champions. Copyright © Restrictions
Exponents
by Ron Kurtus (revised 28 December 2010)
When you multiply a number or variable by itself numerous times, you can designate the result in the exponential notation to make it more convenient to write. It is a form of algebraic shorthand.
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. 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. 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 xy. Thus, if we later say that x = 2 and y = 3, we would have xy = 23 = 8.
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.
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.
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 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
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