Rules with Exponents

Rules with Exponents

Objective: simplify expression with exponents​ using exponential rules.

Fill in the Blanks

WARM-UP: What does 2³ equal?

​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

Fill in the Blanks

m⁷ x m¹¹ = m^?

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

​=

Multiple Choice

Simplify: a³b⁴ x a⁶b²

• a⁹b⁶

• a⁹b¹⁰

• a¹⁸b⁸

• a⁹b⁸

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)

Multiple Choice

$$2n^4\cdot6n^3=$$  

• 8n⁴
• 12n⁷

• 8n¹⁶
• 12n¹²

• 8n⁷

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

Fill in the Blanks

Simplify (p⁴)³ = p^?

​

​(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 =

​=

Multiple Choice

Simplify: (2w³)⁵

• 32w¹⁵

• 10w¹⁵

• 10w⁸

• 32w⁸

Multiple Choice

Simplify: (3x^-2y⁹)³

• 27x^-6y²⁷

• 9x^-6y²⁷

• 9x¹y¹²

• 27x¹y¹²

QUOTIENT RULE

​EXAMPLE

Fill in the Blanks

$\frac{h^{32}}{h^{13}}=h^?$

QUOTIENT RULE

EXAMPLE​​

Fill in the Blanks

$\frac{m^4}{m^{-2}}=m^?$  

Fill in the Blanks

What is the value of m?

$\frac{k^m}{k^7}=k^{15}$  

QUOTIENT/NEGATIVE RULE

EXAMPLES​​

Fill in the Blanks

Simplify: $\frac{6^{-2}}{2^{-3}}$  

(make sure you simplify the fraction)

QUOTIENT/NEGATIVE RULE

EXAMPLE​​

Multiple Choice

Simplify: $$\frac{x^3y^{-3}}{z^{-3}}$$  

• $$x^3y^3z^3$$  

• $$\frac{y^3}{x^3z^3}$$  

• $$\frac{x^3z^3}{y^3}$$  

• $$\frac{x^3y^3}{z^3}$$  

• $$\frac{1}{x^3y^3z^3}$$  

QUOTIENT/NEGATIVE RULE

EXAMPLE​​

Multiple Choice

Simplify: $$\frac{10h^9}{5h^7}$$  

• $$\frac{2}{h^2}$$  

• $$\frac{1}{2h^2}$$  

• $$15h^{16}$$  

• $$2h^2$$  

• $$5h^2$$  

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

Multiple Choice

Simplify:  $$\left(\frac{a}{b}\right)^{-3}$$  

• $$\frac{b^3}{a^3}$$  

• $$\frac{b^3}{a}$$  

• $$\frac{-3a}{-3b}$$  

• $$\frac{3b}{3a}$$  

• $$\frac{a^3}{b^3}$$  

Multiple Choice

Simplify:  $$\left(\frac{4}{9}\right)^{-3}$$  

• $$\frac{729}{64}$$  

• $$\frac{64}{729}$$  

• $$\frac{12}{27}$$  

• $$\frac{27}{12}$$  

• $$-\frac{729}{64}$$  

Fill in the Blanks

Simplify:  $2^{-5}$  

Fill in the Blanks

Simplify:  $\left(\frac{1}{4}\right)^{-3}$  

Multiple Choice

Simplify:  $$\left(\frac{w^4}{2}\right)^{-5}$$  

• $$\frac{32}{w^{20}}$$  

• $$\frac{w^{20}}{32}$$  

• $$\frac{10}{w^{20}}$$  

• $$\frac{10}{w^9}$$  

• $$\frac{32}{w^9}$$