An integer is a number with no fractional part that includes the counting numbers {1, 2, 3, 4, …} , zero {0} and the negative of the counting numbers {- 2, -1, 0, 1, 2}. An exponent of a number says how many times to use that number in a multiplication.
Let's start by reviewing the rules for exponents

I. Multiplying

When you multiply same bases you add exponents.

x⁴ •x⁵ = x⁴⁺⁵ = x⁹

What if an exponent is negative? Same thing add exponents.

x⁶ •x^-4 = x^6+(-4) = x²

What if there is more than one variable? Do each base separately.

(xy⁶)(x³y⁴) = x¹⁺³ y⁶⁺⁴ = x⁴ y¹⁰

What if there is a coefficient in front of the variable?

3x² •-2x³ =

(3 •-2)•(x² • x³) =   Use the commutative property to rearrange
-6x⁵   multiply the coefficients and add exponents

II. Dividing

When you divide same bases you subtract exponents



What if there is more than one variable? Do each base separately.



What if there is a coefficient in front of the variable? Divide the coefficients.



What if the exponent is negative?



III. Raising a power to a power

When you raise a power to a power you multiply exponents.

(x³)⁵ = x^3•5 = x¹⁵

What if there is more than one variable?

(x²y)³ = x^2•3 y^1•3 = x⁶y³

What if there is a coefficient?

(2x⁴y²)⁴ = 2⁴ x^4•4y^2•4 = 16x¹⁶y⁸



IV. Negative exponent Rule   


2nd change floors if the exponent is "unhappy"



Let's look at some more challenging examples



Remember to work slowly and meticulously. You will need to memorize the rules for exponents. A shortened version:

Multiply → Add exponents

Divide → Subtract exponents

Power to a power → Multiply exponents

Negative → Change "floors"

Related Links: Math algebra Exponents Exponents and Powers The Laws of Exponents Integers as Exponents Evaluate Exponents Evaluate integers with exponents Rational (Fractional) Exponents Rational (Fractional) Exponents Zero Exponents