An exponent is shorthand for representing how many times a number or variable is multiplied by itself.

Why use exponents:

We can use exponents to represent very large numbers or very small numbers.

Integers raised to a positive exponent:

Suppose we want 2 times itself 3 times, 2 * 2 * 2 We can use exponents as follows: 2^{3} where 2 is the base and 3 is the exponent

Both of these expressions equal each other 2^{3} = 2 * 2 * 2

Variables raised to a positive exponent:

Suppose we want x times itself 3 times, x * x * x We can use exponents as follows: x^{3} where x is the base and 3 is the exponent

Both of these expressions equal each other x^{3} = x * x * x

Product Rule for Exponents:

Given a number a with exponents m and n, we have: a^{m} * a^{n} = a^{m + n}

Product Rule for Exponents Example:

Let a = 5 with exponents m = 4 and n = 3, we have: 5^{4} * 5^{3} = 5^{4 + 3} 5^{4} * 5^{3} = 5^{7} 625 * 125 ? 78125 78,125 = 78,125

Quotient Rule for Exponents:

Given a number a with exponents m and n, we have:

a^{m - n} =

a^{m}

a^{n}

Quotient Rule for Exponents Example:

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

5^{4 - 3} =

5^{4}

5^{3}

5^{1} =

625

125

5 = 5

Power of a Power Rule for Exponents:

Given a number a with exponents m and n, we have: (a^{m})^{n} = a^{mn}

Power of a Power Rule for Exponents Example:

Let a = 5 with exponents m = 4 and n = 3, we have: (5^{4})^{3} = 5^{4 * 3} 625^{3} = 5^{12} 244,140,625 = 244,140,625

Power of a Product Rule for Exponents:

Given a number a and a number b with exponent n, we have: (ab)^{n} = a^{n}b^{n}

Power of a Product Rule for Exponents Example:

Let a = 4 and b = 3 and n = 5, we have: (4 * 3)^{5} = 4^{5}3^{5} 12^{5} = 1024 * 243 248832 = 248832

Power of a Quotient Rule for Exponents:

Given a number a and a number b with exponent n, we have: (a/b)^{n} = a^{n}/b^{n}

Power of a Quotient Rule for Exponents Example:

Let a = 4 and b = 3 and n = 5, we have: (4/3)^{5} = 4^{5}/3^{5} (1.3333333333333)^{5} = 1024/243 4.21399176955 = 4.21399176955

Negative Integer Exponent Rule:

Given a number a with exponent -n, we have: a^{-n} = (1/a)^{n}

Zero Exponent Rule:

Given a number a with exponent 0, any we have: a^{0} = 1 Any number or integer raised to the 0 power (0 as an exponent) equals 1

How does the What is an Exponent Calculator work?

This lesson walks you through what an exponent is, the product rule for exponents, the quotient rule for exponents, the 0 power rule, the power of a power rule for exponents

What 5 formulas are used for the What is an Exponent Calculator?