Algebra Properties of Exponents

​Algebra 1: Properties of Exponents Lesson

​In this lesson, we will go over the properties of exponents and you will have an opportunity to demonstrate your knowledge with some practice questions. Get your journal out, take notes, and attempt the practice questions.

​Product of Powers

​Rule:

When you have the same base, you are allowed to add the exponents.

Multiple Choice

Simplify:

$$\left(x^6\right)\left(x^5\right)$$  

• $$2x^{11}$$  

• $$2x^{30}$$  

• $$x^{11}$$  

• $$x^{30}$$  

Match

Match the following:

• $$x^3x^4$$

• $$x^{-3}x^{12}$$

• $$x^3x^3$$

• $$x^8x^4$$

• $$x^{-1}x^6$$

• $$x^7$$

• $$x^9$$

• $$x^6$$

• $$x^{12}$$

• $$x^5$$

Math Response

Simplify

$$y^2\cdot y^7$$  

Type answer here

Correct Answer: y^9

Deg^°

Rad

​Quotient of Powers

​Rule:

When you have the same base, you are allowed to subtract the exponents.

Match

Match the following

• $$\frac{b^6}{b^{-2}}$$

• $$\frac{b^{10}}{b^5}$$

• $$\frac{b^{12}}{b^{15}}$$

• $$\frac{b^7}{b^4}$$

• $$\frac{b}{b^6}$$

• $$b^8$$

• $$b^5$$

• $$b^{-3}=\frac{1}{b^3}$$

• $$b^3$$

• $$b^{-5}=\frac{1}{b^5}$$

Math Response

Simplify:

$$\frac{x^2y^8}{xy^2}$$

Type answer here

Correct Answer: xy^6

Deg^°

Rad

Math Response

Simplify:

$$\frac{x^3y^8}{x^5y^{10}}$$

Type answer here

Correct Answer: \frac{1}{x^2y^2}

Deg^°

Rad

​Power of a Power

​Rule:

When you have the same base, you are allowed to multiply the exponents.

Match

Match the following

• $$\left(x^2\right)^7$$

• $$\left(x^3\right)^3$$

• $$\left(\frac{2x}{3y}\right)^3$$

• $$\left(\frac{4x}{3y^2}\right)^2$$

• $$\left(2x^2y^4\right)^3$$

• $$x^{14}$$

• $$x^9$$

• $$\frac{8x^3}{27y^3}$$

• $$\frac{16x^2}{9y^4}$$

• $$8x^6y^{12}$$

Math Response

Simplify the following:

$$\left(2x^2y^3z\right)^3$$

Type answer here

Correct Answer: 8x^6y^9z^3

Deg^°

Rad

Math Response

Simplify the following:

$$\left(3x^5y^3z^{-2}\right)^2$$

Type answer here

Correct Answer: \frac{9x^{10}y^6}{z^4}

Deg^°

Rad

​Review of the Properties of Exponents

​More examples:

​On the following slides, you will demonstrate your knowledge over the properties of exponents. You may use your notes as you attempt each problem.

Multiple Choice

$$\left(\frac{2r}{3s}\right)\left(\frac{9s}{4r^2}\right)=$$  

• $$6rs$$  

• $$\frac{3s}{2r}$$  

• $$\frac{3}{2r}$$  

• $$\frac{8r^3}{27s^2}$$  

Multiple Choice

Which of the following is equivalent to the given expression?

$$\left(\frac{6}{x}\right)^2$$  

• $$\frac{12}{x}$$  

• $$\frac{12}{x^2}$$  

• $$\frac{36}{x}$$  

• $$\frac{36}{x^2}$$  

Match

Match the following:

• $$x^3x^4$$   

• $$\left(x^3\right)^4$$  

• $$\left(xy\right)^4$$  

• $$\frac{x^6}{x^4}$$  

• $$\left(\frac{x}{y}\right)^4$$

• $$x^7$$

• $$x^{12}$$

• $$x^4y^4$$

• $$x^2$$

• $$\frac{x^4}{y^4}$$

Match

Match the following

• $$\left(\frac{x}{y}\right)^3$$  

• $$\frac{1}{x^3}$$  

• $$x^0$$  

• $$\left(\frac{x}{y}\right)^{-3}$$

• $$\frac{x^3}{y^3}$$

• $$x^{-3}$$

• $$1$$

• $$\frac{y^3}{x^3}$$

Math Response

Simplify

$$\left(3x^9\right)\left(xy^{11}\right)$$  

Type answer here

Correct Answer: 3x^{10}y^{11}

Deg^°

Rad

Multiple Choice

Simplify

$$\left(x^8\right)^7$$  

• $$x^{56}$$  

• $$x^{15}$$  

• $$x^{ }$$  

• $$x^2$$  

Math Response

Simplify

$$\left(x^3\right)^{10}$$  

Type answer here

Correct Answer: x^{30}

Deg^°

Rad

Multiple Choice

Simplify

$$\left(cd\right)^{11}$$  

• $$cd$$  

• $$c^{11}d^{11}$$  

• $$cd^{11}$$  

• $$c^1d^1$$  

Math Response

Simplify

$$\left(xy\right)^7$$  

Type answer here

Correct Answer: x^7y^7

Deg^°

Rad

Multiple Choice

Simplify

$$\frac{y^7}{y^2}$$  

• $$y^5$$  

• $$y^9$$  

• $$y^{14}$$  

• $$y$$  

Math Response

Simplify

$$\frac{z^{10}}{z^3}$$  

Type answer here

Correct Answer: z^7

Deg^°

Rad

Multiple Choice

Simplify

$$\left(\frac{m}{n}\right)^4$$  

• $$\frac{m^4}{n^4}$$  

• $$\frac{4m}{n}$$  

• $$\frac{m^4}{n}$$  

• $$\frac{m^4}{n^{-4}}$$  

Math Response

Simplify

$$\left(\frac{x}{y}\right)^7$$  

Type answer here

Correct Answer: \frac{x^7}{y^7}

Deg^°

Rad

Multiple Choice

Simplify

$$x^{-3}$$  

• $$\frac{1}{x^3}$$  

• $$x^3$$  

• $$\frac{1}{x^{-3}}$$  

• $$x$$  

Math Response

Simplify

$$\frac{1}{y^{-3}}$$  

Type answer here

Correct Answer: y^3

Deg^°

Rad

Multiple Choice

Which expression is equivalent to   $$\frac{x^{-5}y^4}{x^2y}$$ for all values of x and y where the expression is defined?

• $$\frac{y^4}{x^{-7}}$$

• $$x^7y^3$$

• $$\frac{y^3}{x^7}$$

• $$\frac{y^4}{x^7}$$

Math Response

Write a simplified expression for

  $$\frac{x^{-8}y^6}{x^{ }y^3}$$

for all values of x and y where the expression is defined?

Make sure all the exponents are positive.

Type answer here

Correct Answer: \frac{y^3}{x^9}

Deg^°

Rad

Math Response

Write a simplified expression for

  $$\left(\frac{2x^2}{3y^3}\right)^3$$

for all values of x and y where the expression is defined?

Make sure all the exponents are positive.

Type answer here

Correct Answer: \frac{8x^6}{27y^9}

Deg^°

Rad

Math Response

In the expression shown,  $$x$$  is a value where the expression is defined.

 

$$(2x^5)(3x^{-2})^2$$

 

What is the simplest form of the expression? Write your answer in the form $$ax^b$$ .

Enter your answer in the box provided.

Type answer here

Correct Answer: 18x

Deg^°

Rad

Math Response

Write a simplified expression that is equivalent to

$$\left(3xy^2\right)\left(4xy\right)$$

Type answer here

Correct Answer: 12x^2y^3

Deg^°

Rad