​EXPONENTS

​2³

​base

​exponent

​2³ means 2 x 2 x 2 which equals 8

Special Case Exponents

Any BASE to the power of 1 simply equals the BASE.

​Example: 6¹ =

Example: y¹=

​​

If there is no exponent, there is an invisible 1​

Example: x = x¹

Example: 45 = 45¹ ​

Special Case Exponents

​ZERO RULE

​

Any BASE to the power of 0 simply equals ONE.

​

Example: 4⁰ =

Example: z⁰=

PRODUCT RULE

EXAMPLES

a² ⋅ a⁵ = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a⁷

​

SHORTCUT: just ADD the EXPONENTS

​y⁶ ⋅ y⁷ =​

​a^x ⋅ a^y = a^x+y

Multiplying with the Multiple Bases

EXAMPLE

​z⁵y³x^-2 ⋅ z⁶y⁵x⁷ = z⁵ ⋅ y³ ⋅​ x^-2 ⋅​ z⁶ ⋅ y⁵ ⋅​ x⁷

PRODUCT RULE

​= z⁵ ⋅ z⁶ ⋅ y³ ⋅ y⁵ ⋅​ x^-2 ⋅​ x⁷

​=

EXAMPLE

​4x³ ⋅ 5x² = 4 ⋅ x³ ⋅ 5 ⋅ x²

PRODUCT RULE

​Multiplication is commutative, so we can rearrange them.

​= 4 ⋅ 5 ⋅ x³ ⋅ x²

​=

Multiplying with Coefficients (that are not 1)

POWER RULE

EXAMPLES

(a³)² = (a ⋅ a ⋅ a)² = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a⁶

​

​​SHORTCUT: just MULTIPLY the EXPONENTS

(w⁷)⁴​ =

​(a^x)^y = a^x⋅y

​

​(a³b⁶c^-2)² = a^3⋅2 ⋅ b^6⋅2⋅ c^-2⋅2

POWER RULE

​=

With Coefficients that are not 1

​(4x³y⁴)² = (4¹x³y⁴)²

​=

With multiple variables

​(a³b⁶c^-2)⁰ =

POWER RULE

​=

With exponents that are ZERO

​(3x⁰y^-5)³ =

​=

QUOTIENT RULE

​EXAMPLE

QUOTIENT RULE

EXAMPLE​​

QUOTIENT/NEGATIVE RULE

EXAMPLES​​

QUOTIENT/NEGATIVE RULE

EXAMPLE​​

QUOTIENT/NEGATIVE RULE

EXAMPLE​​

PRE-NEGATIVE RULE

​Reciprocal: simply a fraction flipped

​EXAMPLES: What is the reciprocal of each?

NEGATIVE RULE

When an expression is raised to a negative power you apply the positive exponent to the reciprocal.

​

STEP 1: Flip the expression and change the exponent to positive

STEP 2: Apply the positive exponent

NEGATIVE RULE

STEP 1: Flip the expression and change the exponent to positive

STEP 2: Apply the positive exponent

​EXAMPLES