Product & Quotient Rules of Exponents

Product & Quotient Rules of Exponents

Laws of Exponents

Product Rule

• Look at the right and see what happens when you multiply powers that have the same base

• When you expand 3² you have two 3s being multiplied and when you expand 3³ you have three 3s being multiplied.

• All together how many 3s are you multiplying?

Product Rule Short Cut?!!

When you multiply same base powers you keep the base and add the exponents.

x² * x⁷ = x²⁺⁷ = x⁹

Multiple Choice

Simplify the following:

$$a^{11}\cdot a^2\cdot a^0$$  

• $$a^{13}$$  

• $$1$$  

• $$a^{22}$$  

• $$a^0$$  

Multiple Choice

Simplify the following:

$$x^3\cdot x\cdot x^{-2}$$  

• $$x^1$$  

• $$x^5$$  

• $$x^6$$  

• $$x^2$$  

What if we have a more than one variable or coefficients?

When you multiply same base powers you multiply the coefficients and combine like bases by adding the exponents.

(2y⁵x²)(6y^-3x⁷) = 2⋅6 y^5+(-3) x²⁺⁷ = 12y²x⁹

Multiple Choice

Simplify the following:

$$\left(4a^4b^5\right)\left(7a^{-3}bc^7\right)$$

• $$28abc^{13}$$

• $$28ab^6c^7$$

• $$11ab^6c^7$$

• $$11abc^{13}$$

Quotient Rule

• When dividing powers that have the same base think of dividing as the inverse of multiplying, so what is the inverse of adding?

• When you expand x⁵ you have five Xs being multiplied in the numerator and when you expand x² you have two Xs being multiplied in the denominator.

• So what happens when you have the Xs in the numerator and Xs in the denominator? Yup! you cross out the matching pairs!

• Now what did you end up with?

Quotient Rule Short Cut?!!

When you divide same base powers you keep the base and subtract the exponents.

x⁷ / x² = x^7-2 = x⁵

Multiple Choice

Simplify the following:

$$\frac{x^{16}}{x^2}$$  

• $$x^{14}$$  

• $$x^8$$  

• $$x^{18}$$  

• $$x^{-8}$$  

Multiple Choice

Simplify the following:

$$\frac{x^9y^7}{x^2y^6}$$  

• $$xy^8$$  

• $$\left(xy\right)^{24}$$  

• $$x^7y$$  

• $$x^{11}y^1$$  

Multiple Choice

Simplify the following:

$$\frac{x^7}{x^{-4}}$$  

• $$x^{11}$$  

• $$x^3$$  

• $$x^{-3}$$  

• $$x^{-11}$$  

Multiple Choice

Simplify the following:

$$\frac{x^3}{x^8}$$  

• $$x^{-5}$$  

• $$\frac{1}{x^5}$$  
  

• $$x^{-3}$$  

• $$\frac{1}{x^{11}}$$  

What if we have coefficients?

Multiple Choice

Simplify the following:

$$\frac{14a^4b^{-3}}{2a^2b^{-7}}$$

• $$7a^2b^4$$

• $$12a^2b^4$$

• $$7a^2b^{-10}$$

• $$12a^2b^{-10}$$

Multiple Choice

Simplify the following:

$$\frac{8a^5b^{-3}}{16a^{-2}b}$$

• $$\frac{1}{2}a^3b^{-2}$$

• $$2a^7b^{-4}$$

• $$2a^3b^{-2}$$

• $$\frac{1}{2}a^7b^{-4}$$

Poll

How is your understanding of this topic?