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, 53 = (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 53 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, 53 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) = x3y2
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:
n3 + 3n3 = 4n3
6a4 – 2a4 = 4a4
2x3y2 + 4x3y2 = 6x3y2
Zero Exponent Rule
One helpful exponent rule is that any non-zero number raised to the zero power equals 1:
a0 = 1
So, no matter how complicated the base is, if you raise it to the zero power, it equals 1. For example:
(62x5y3)0 = 1
Knowing this rule can save you a lot of pointless calculation!
However, if the base is 0, matters become complicated. 00 has an indeterminate form.
Product Rule and Quotient Rule
When you multiply exponents with the same base, keep the base add the exponents:
aman = am+n
(53)(52) = 53+2 = 55
Similarly, divide exponents with the same base by keeping the base and subtracting the exponents:
am/an = am-n
53/52 = 53-2 = 51 = 5
x-3/x2 = 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 = ambm
(3×2)2 = (32)(22) = 9×4 = 36
(x2y2)3 = x6y6
Power of a Quotient
Distribution works when dividing numbers, too. Distribute the exponent to all values within the brackets:
(a/b)m = am/bm
(4/2)2 = 42/22 = 16/4 = 4
(4x3/5y4)2 = 42x6/52y8 = 16x6/25y8
Power of a Power Exponent Rule
When raising a power by another power, keep the base and multiply the exponents together:
(am)n = amn
(23)2 = 23×2 = 26
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/am
2-2 = 1/22 = 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:
am/n = (n√a)m
33/2 = (2√3)3 which is about 5.196
Check your math, since you know 33/2 = 31.5. Note this is not the same as 2√33, 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