What is an Exponent

What is an Exponent Definition:

Shorthand for how many times
a number or variable
is multiplied by itself.

Why use exponents:

Represent large numbers or small numbers.

Integers raised to a positive exponent:

2 times itself 3 times, 2 * 2 * 2
We can use exponents as follows:
2³ where 2 is the base and 3 is the exponent

Both of these expressions equal each other
2³ = 2 * 2 * 2

Variables raised to a positive exponent:

x times itself 3 times, x * x * x
We can use exponents as follows:
x³ where x is the base and 3 is the exponent

Both of these expressions equal each other
x³ = x * x * x

Product Rule for Exponents:

Given a number a with exponents m and n:
a^m * aⁿ = a^{m + n}

Product Rule for Exponents Example:

Let a = 5 with exponents m = 4 and n = 3:
5⁴ * 5³ = 5^{4 + 3}
5⁴ * 5³ = 5⁷
625 * 125 ? 78125
78,125 = 78,125

Quotient Rule for Exponents:

Given a number a with exponents m and n:

am - n  =	am
	an

Quotient Rule for Exponents Example:

Let a = 5 with exponents m = 4 and n = 3:

54 - 3  =	54
	53

51  =	625
	125

5 = 5

Power of a Power Rule for Exponents:

Given a number a with exponents m and n:
(a^m)ⁿ = a^mn

Power of a Power Rule for Exponents Example:

Let a = 5 with exponents m = 4 and n = 3:
(5⁴)³ = 5^{4 * 3}
625³ = 5¹²
244,140,625 = 244,140,625

Power of a Product Rule for Exponents:

Given a number a and a number b with exponent n:
(ab)ⁿ = aⁿbⁿ

Power of a Product Rule for Exponents Example:

Let a = 4 and b = 3 and n = 5:
(4 * 3)⁵ = 4⁵3⁵
12⁵ = 1024 * 243
248832 = 248832

Power of a Quotient Rule for Exponents:

Given a number a and a number b with exponent n:
(a/b)ⁿ = aⁿ/bⁿ

Power of a Quotient Rule for Exponents Example:

Let a = 4 and b = 3 and n = 5:
(4/3)⁵ = 4⁵/3⁵
(1.3333333333333)⁵ = 1024/243
4.21399176955 = 4.21399176955

Negative Integer Exponent Rule:

Given a number a with exponent -n:
a^-n = (1/a)ⁿ

Zero Exponent Rule:

Given a number a with exponent 0, any we have:
a⁰ = 1
Anything raised to the 0 power = 1