An **exponent** or **power** is a superscript over a number (the base) that tells how many times you multiple that number by itself. It is a shorthand for repeated multiplication that makes writing equations simpler.

### Reading and Writing Exponents

For example, 5^{3} = (5)(5)(5) = 125. Here, the number 5 is the **base** and the number 3 is the **exponent** or **power**. You can read the expression 5^{3} as “five raised to the third power” or “five raised to the power of three.” However, a number raised to the power of 3 is generally read as “cubed”. So, 5^{3} is “five cubed.” A number raised to the power of 2 is “squared.”

Many time, exponents combine with algebra. For instance, here is an expanded form and exponential form of an equation using *x* and *y*:

(x)(x)(x)(y)(y) = x^{3}y^{2}

### Exponent Rules and Examples

Exponents simplify writing extremely large or very small numbers. This is why they find use in scientific notation. Understanding the rules for exponents makes working with them much easier.

#### Addition and Subtraction

You can add and subtract numbers with exponents, but only when the base and exponent of the terms are the same. For example:

n^{3} + 3n^{3} = 4n^{3}

6a^{4} – 2a^{4} = 4a^{4}

2x^{3}y^{2} + 4x^{3}y^{2} = 6x^{3}y^{2}

#### Zero Exponent Rule

One helpful exponent rule is that any non-zero number raised to the zero power equals 1:

a^{0} = 1

So, no matter how complicated the base is, if you raise it to the zero power, it equals 1. For example:

(6^{2}x^{5}y^{3})^{0} = 1

Knowing this rule can save you a lot of pointless calculation!

However, if the base is 0, matters become complicated. 0^{0} has an indeterminate form.

#### Product Rule and Quotient Rule

When you multiply exponents with the same base, keep the base add the exponents:

a^{m}a^{n} = a^{m+n}

(5^{3})(5^{2}) = 5^{3+2} = 5^{5}

Similarly, divide exponents with the same base by keeping the base and subtracting the exponents:

a^{m}/a^{n} = a^{m-n}

5^{3}/5^{2} = 5^{3-2} = 5^{1} = 5

x^{-3}/x^{2} = x^{(-3-2)} = x^{-5}

#### Power of a Product

Another way of expressing a base multiplied by an exponent is distributing the exponent to each base:

(ab)^{m} = a^{m}b^{m}

(3×2)^{2} = (3^{2})(2^{2}) = 9×4 = 36

(x^{2}y^{2})^{3} = x^{6}y^{6}

#### Power of a Quotient

Distribution works when dividing numbers, too. Distribute the exponent to all values within the brackets:

(a/b)^{m} = a^{m}/b^{m}

(4/2)^{2} = 4^{2}/2^{2} = 16/4 = 4

(4x^{3}/5y^{4})^{2} = 4^{2}x^{6}/5^{2}y^{8} = 16x^{6}/25y^{8}

#### Power of a Power Exponent Rule

When raising a power by another power, keep the base and multiply the exponents together:

(a^{m})^{n} = a^{mn}

(2^{3})^{2} = 2^{3×2} = 2^{6}

#### Negative Exponent Rule

When raising a number to a negative exponent, use the reciprocal of the base and make the exponent sign positive:

a^{-m} = 1/a^{m}

2^{-2} = 1/2^{2} = 1/4

#### Fractional Exponent

Another way of writing a base raised to a fraction is to take the denominator root of the base and raise it to the numerator power:

a^{m/n} = (^{n}**√**a)^{m}

3^{3/2} = (^{2}**√**3)^{3} which is about 5.196

Check your math, since you know 3^{3/2} = 3^{1.5}. Note this is *not* the same as ^{2}**√**3^{3}, which equals 3. Brackets are everything!

### References

- Hass, Joel R.; Heil, Christopher E.; Weir, Maurice D.; Thomas, George B. (2018).
*Thomas’ Calculus*(14th ed.). Pearson. ISBN 9780134439020. - Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010).
*NIST Handbook of Mathematical Functions*. National Institute of Standards and Technology (NIST), U.S. Department of Commerce, Cambridge University Press. ISBN 978-0-521-19225-5. - Rotman, Joseph J. (2015).
*Advanced Modern Algebra, Part 1*. Graduate Studies in Mathematics. Vol. 165 (3rd ed.). Providence, RI: American Mathematical Society. ISBN 978-1-4704-1554-9. - Zeidler, Eberhard; Schwarz, Hans Rudolf; et al. (2013) [2012]. Zeidler, Eberhard (ed.).
*Springer-Handbuch der Mathematik I*(in German). Vol. I (1 ed.). Berlin / Heidelberg, Germany: Springer Spektrum, Springer Fachmedien Wiesbaden. doi:10.1007/978-3-658-00285-5