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# 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

7by itself4times, you would get

7 × 7 × 7 × 7 = 7, where^{4}7is thebaseand4is theexponent. 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 **5 ^{6}**, where

**5**is the

**base number**and

**6**is the

**exponent**. It is also called

*.*

**5**raised to the**6th**powerCertainly, **5 ^{6}** is more convenient to write than

**15,625**.

You can also use exponents with decimal numbers.

For example ** 1.3 × 1.3 = 1.3 ^{2}** and

**0.2 × 0.2 × 0.2 = 0.2**.

^{3}## 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 **x ^{4}**. In this case,

**x**is the base and

**4**is the exponent.

Notethat we used*to denote multiplication instead of×, since that multiplication sign can be confused with the letterx. Many algebra books use·as multiplication, but that also can be confused with the decimal point, such as3·5versus3.5. 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 **x ^{z}**. Thus, if we later assign values to the variables, such as

**x = 2**and

**z = 3**, we would have

**x**.

^{z}= 2^{3}= 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, **x ^{4}** can be called

**x**

*raised to the*. Likewise,

**4**th power**x**is

^{10}**x**

*raised to the*. The same is true for numbers, where

**10**th power**5**is

^{6}*.*

**5**raised to the**6**th power

Notethat sometimes on the Internet, you will see the^sign to indicate raised to a power. For example,b^5 = band^{5}10^3 = 10.^{3}

### 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, **x ^{2}** is usually called

**x-squared**, since the area of a square is

**x*x**.

Following that logic, **x ^{3}** is usually called

**x-cubed**, since the volume of a cube is

**x*x*x**.

Likewise, **5 ^{2}** is

**5-squared**and

**5**is

^{3}**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

### Websites

**Exponents: Basic Rules** - PurpleMath.com

**Exponent Rules** - RapidTables.com

**Laws of Exponents** - MathisFun.com

**Exponents Calculator** - CalculatorSoup.com

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Where are you now?

## Exponents in Algebra