An exponent or otherwise called power is a superscript over a base number that tells how many times you multiply that number by itself. It is a short version of repeated multiplication that makes writing equations simpler.
Exponent Rules
Zero Rule–a0 = 1
Product Rule-am x an = am+n
Quotient Rule-am/an = am-n
Power of a Product–(ab)m = ambm
Power of a Quotient–(a/b)m = am/bm
Power of a Power–(am)n = amn
Negative Exponent–a-m = 1/am
Fractional Exponent–am/n = (n√a)m
Check out 1 to the power of 94
Reading and Writing Exponents
For instance, 53 = (5)(5)(5) = 125. In this example, the number 5 is the base and the number 3 is the exponent (power). You can read the expression 53 as “five raised to the power of three” or “five raised to the third power.” However, a number raised to the power of 3 is typically read as “cubed”. Accordingly, 53 is “five cubed.” A number raised to the power of 2 is “squared.”
Many times, exponents combine with algebra. For example, here is an extended form and exponential form of an equation using x and y:
(x)(x)(x)(y)(y) = x3y2
Exponent Rules and Examples
Exponents facilitate writing very small or extremely large numbers. This is the reason why they find use in scientific notation. Understanding the rules for exponents makes working with them much more effortless.
Addition and Subtraction
Only when the base and exponent of the terms are the same you can add and subtract numbers with exponents. For instance:
n3 + 3n3 = 4n3
6a4 – 2a4 = 4a4
2x3y2 + 4x3y2 = 6x3y2
Zero Exponent Rule
The practical exponent rule is that any non-zero number raised to the zero power equals 1:
a0 = 1
So, no matter how complex the base is, if you raise it to zero power, it equals 1. For instance:
(62x5y3)0 = 1
Understanding this rule can save you a lot of pointless calculations!
Yet, if the base is 0, matters become complex. 00 has an indeterminate form.
Product Rule and Quotient Rule
Keep the base and add the exponents when you multiply exponents with the same base:
aman = am+n
(53)(52) = 53+2 = 55
Likewise, divide exponents with the same base by holding 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
Distributing the exponent to each base is another way of expressing a base multiplied by an exponent:
(ab)m = ambm
(3×2)2 = (32)(22) = 9×4 = 36
(x2y2)3 = x6y6
Power of a Quotient
Distribution functions when dividing numbers, as well. Distribute the exponent to all values in 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
Keep the base and multiply the exponents together when raising a power by another power:
(am)n = amn
(23)2 = 23×2 = 26
Negative Exponent Rule
Use the reciprocal of the base and make the exponent sign positive when raising a number to a negative exponent:
a-m = 1/am
2-2 = 1/22 = 1/4
Fractional Exponent
Another method 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 are aware that 33/2 = 31.5. Remember that brackets are everything, so, note this is not the same as 2√33, which equals 3.
Read about How To Use An Exponential Calculator
