Simplifying with Exponents

Definition: x^a/x^b = x ^a^-b

When this works: When the bases are the same and the monomials are dividing each other.

Situations when this works:

a¹⁵/a³⁶
x² ÷ x⁵

Notice that the bases are the same in both examples.

1. Example: Quotient of Powers with Constants

Simplify 4⁵/4³

Rewrite 4⁵/4³

4*4*4*4*4/ 4*4*4

Cancel the 4’s in the numerator and denominator

4*4*4*4/ 4*4*4

How many 4’s remain? 2

4 *4 = 4² = 16

Simplify using the Quotient of Powers

4⁵/4³
4^5-3
4²

2. Example: Quotient of Powers with Variables

Simplify p⁶/p².

Before you are tempted to answer p³, return to the Quotient of Powers Rule:

x^a/x^b = x^a^-b
p^6-2
p⁴

Why Does This Make Sense?

1. Rewrite p⁶/p²

p*p*p*p*p*p/ p*p

2. Cancel the p's in the numerator and denominator.

*>*p*p*p*p/

3. How many p's remain? 4

p*p*p*p = p⁴

3. Example: Quotient of Powers with Coefficients Other Than 1 and 0

Simplify 4y⁵ ÷ 24y⁴

1. Break down the exercise into pieces. Match constants with constants and variables with variables.
4/24 * y⁵/y⁴

2. Simplify each piece.
1/6 * y^5-41/6 * y1

3. Put the pieces back together.
1/6 * y
y/6

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