# How to use the POWER function

**What is the POWER function?**

The POWER function calculates a number raised to a power. You can also use the caret ^ character to perform a exponentiation in an Excel formula.

## 1. Introduction

**What is exponentiation?**

Exponentiation is a mathematical operation involving exponents. It means raising a number, known as the base, to a specified power. The exponent indicates how many times to multiply the base by itself.

base^{exponent}

Roots are the inverse operation, giving the base when the power and number are known. For example,

x^{5} = 3125

3125^{(1/5)} = 5

x = 5

5^{5} = 3125

This makes it possible to calculate roots using the POWER function, however, to calculate square roots you can also use the SQRT function.

**How are roots and logarithms related?**

Roots and logarithms are inverse mathematical operations:

A root "undoes" exponentiation, finding the base number when given the power and result.

For example, âˆš16 = 4, since 4^{2} = 16. The root reveals the base of 4. âˆš16 = 16^{(1/2)}

A logarithm is the inverse of exponentiation as well, returning the exponent that a base must be raised to in order to produce a number.

For example, =log(8,2) = 3, since 2^{3} = 8. The logarithm returns the exponent 3.

Logarithms calculate the exponent and roots calculate the base. This example demonstrates how to calculate the exponent:

5^{x} = 3125

LOG5(3125) = 5

x = 5

5^{5} = 3125

**What are the other logarithmic functions in Excel?**

Excel function | Description |
---|---|

LOG | Returns the logarithm with a given base. |

LN | Returns the natural logarithm (base e) of a number |

LOG10 | Returns the base-10 logarithm of a number |

**The corresponding inverse functions for the logarithm functions in Excel:**

Logarithm | Power to |
---|---|

LOG | Arbitrary base, base^{x}Â or POWER(number, power) |

LN | EXP(number) |

LOG10 | 10^{x}Â or POWER(10,x) |

**What are the exponentiation rules?**

Product Rule: x^{m} * x^{n} = x^{(m+n)}

Exponents with the same base are multiplied when multiplying terms.

Power Rule: (x^{m})^{n} = x^{(m*n)}

When raising a power to a power, the exponents are multiplied.

Power of a Product: (ab)^{n} = a^{n} * b^{n}

Exponents distribute to all factors in a product.

Quotient Rule: x^{m} / x^{n} = x^{(m-n)}

Exponents with the same base subtract when dividing terms.

Power of Quotient: (a/b)^{n} = a^{n} / b^{n}

Exponents distribute to the numerator and denominator when raising a quotient.

Any base raised to the 0 power equals 1.

Negative Exponent Rule: x^{-n} = 1 / x^{n}

## 2. Syntax

POWER(*number, power*)

number |
Required. The number you want toÂ raise. |

power |
Required. The exponent to which the number is raised. |

## 3. Example 1

The first example has the number in cell B3 and the power in cell C3. They are 5 and 2 respectively. The formula is in cell E3:

The result of 5^{2} is 25, here is how:

5^{2 }= 5 * 5 = 25

The second example has the number 2 in cell B4 and the power 8 in cell C4. The formula is in cell E4:

The result of 2^{8} is 256, here is how:

2^{8 }= 2 * 2 * 2* 2 * 2 * 2 * 2 * 2 = 256

The third example has the number in cell B5 and the power in cell C5. They are 10 and 2 respectively. The formula is in cell E5:

The result of 10^{2} is 100, here is how:

10^{2 }= 10 * 10 = 100

The fourth example has the number 10 in cell B6 and the power 6 in cell C6. The formula is in cell E6:

The result of 10^{6} is 256, here is how:

10^{6} = 10 * 10 * 10 * 10 * 10 * 10 = 1,000,000

The fifth example has the number in cell B7 and the power in cell C7. They are 3 and 3 respectively. The formula is in cell E5:

The result of 3^{3} is 27, here is how:

3^{3 }= 3 * 3 * 3 = 9 * 3 = 27

The sixth example has the number 4 in cell B6 and the power 3 in cell C6. The formula is in cell E6:

The result of 4^{3} is 64, here is how:

4^{3} = 4 * 4 * 4 = 64

## 4. Example 2

**What is the area of a circle if the radius is 5 units?**

The area of a circle is found by multiplying the mathematical constant pi (Ï€) by the square of the circle's radius.

A = Ï€ * r^{2}

A = the area of the circle

r = the radius of the circle

Ï€ = pi

Formula in cell C23:

The area is approx. 78.54 square units based on a radius of 5 units. The PI function lets you calculate number pi, it has no arguments. The POWER function takes 5 to the power of 2 which is 25.

Lets plug the values and calculate the area manually. Here is how:

A = Ï€ * r^{2}

A = 3.14159265358979 * 5^{2}

A = 3.14159265358979 * 25

A = 78.5398163397448

## 5. Example 3

**A bacteria population doubles ever 1.5 hour. How many cells will you find after 7 hours if the start population is 50?**

We need to find a pattern in the growth of the bacteria population over time and then use it to calculate the population after 7 hours.

Given information:

- The bacteria population doubles every 1.5 hours.
- The initial population is 50 cells.

The math formula has to be: Population = 50 * 2^{(t/1.5)
}

Formula in cell C21:

The formula returns 1269 cells after 7 hours with a start population of 50. The image above shows a chart illustrating an exponential blue cure, the x-axis title is hours and ranges from 0 to 7 hours. The y-axis title is population and ranges from 0 to 1400 cells.

Go to value 7 on the x-axis, follow an imaginary vertical line upÂ until it intersects with the blue curve. The follow an imaginary horizontal line to the left until you find the y-axis. The y-axis value is near 1300 which corresponds to the calculated value in cell C21.

## 6. Example 4

*If you invest $1,000 today in an account that pays 8% annual interest compounded annually, what will be the future value of your investment after 20 years?*

We need to model a math function that follows the growth of the investment over time and then use it to calculate the amount after 20 years.

Given information:

- The investment compounds annually with an rate of 8%.
- The initial investment is 1,000.

The math formula has to be: Amount= 1000 * 1.08^{y}

Formula in cell C21:

The formula returns 4,660 after 20 years with an initial investment of 1,000. The image above shows a chart illustrating an exponential blue cure, the x-axis represents years and ranges from 0 to 20 years. The y-axis title represents the amount and ranges from 1,000 to 5,000.

Go to year 20 on the x-axis, follow an imaginary vertical line up until it intersects with the blue curve. The follow an imaginary horizontal line to the left until you find the y-axis. The y-axis value is near 4600 which corresponds to the calculated amount in cell C21.

### Comments

The ^ operator is easier to use, example 10^2 = 100, number^power.

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