Properties of Exponents

6.1c- Quotient of Powers, Negative
Exponents, Zero Powers

Obj- To use the rules for Quotient of Powers,
Negative Exponents and Zero Powers

Multiple Choice

Warm-Up #1: Simplify the expression:  c⁴⋅c³=

• ^c12

• c⁴⁺³

• c⁷

Multiple Choice

Warm-Up #2: Simplify the following expression:

(x⁵)⁴

• x⁹

• x²⁰

• x

• x⁵⁴

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:

$$\frac{x^3}{x^8}$$  

• $$x^{-5}$$  

• $$x^5$$  

• $$x^{-3}$$  

• $$\frac{1}{x^{11}}$$  

Negative Exponent Rule

Negative exponents can be written as the reciprocal fraction with a postive exponent

Negative Exponents Examples:

Negative Exponent Rule

Multiple Choice

Simplify x^-7

• -7

• -7x

• $$\frac{1}{x^7}$$

• $$-\frac{1}{x^7}$$

Multiple Choice

Make this negative exponent positive.

9^-3

• 9³

• $$\frac{1}{9^{-3}}$$

• $$\frac{1}{9^3}$$

• 729

Multiple Choice

Rewrite using positive exponents

$$\frac{1}{12^{-4}}$$  

• $$\frac{1}{12^4}$$  

• $$12^{-4}$$  

• $$\frac{1}{20,736}$$  

• $$12^4$$  

Multiple Choice

Rewrite using positive exponents.

3x⁻²

• $$\frac{3}{x^2}$$

• $$\frac{1}{3^2}$$

• $$\frac{x^2}{3}$$

• -3x²

Zero Exponent

• As our exponent increases, we MULTIPLY by the base that many time

• So, every time your exponent decreases, we DIVIDE by the base.

• When the exponent decreases from 1 to zero, we divide the base by itself and end up with 1

Multiple Choice

Any number written to the 0 power is always 1

Example: 8⁰=1

• True

• False

Multiple Choice

x⁰

• 0

• 1

• x

• 12

Multiple Choice

Simplify 10⁰

• 10

• 1

• 0

• 100

Multiple Choice

Simplify -9⁰ 
(Hint: parenthesis matter)

• 1

• -1

• -9

• 9

Extra Examples

Math Response

Answer to #1 $$\frac{x^6}{x^2}$$

Type answer here

Correct Answer: x^4

Deg^°

Rad

Math Response

Answer to #2 $$13xy^0$$

Type answer here

Correct Answer: 13x

Deg^°

Rad

Math Response

Answer #3 $$\left(13xy\right)^0$$

Type answer here

Correct Answer: 1

Deg^°

Rad

Extra Examples

Math Response

Answer to #4: $$2^{-4}$$

(Make the exponent positive)

Type answer here

Correct Answer: \frac{1}{16}

Deg^°

Rad

Math Response

Answer to #5: $$\left(-3\right)^{-2}$$

Type answer here

Correct Answer: \frac{1}{9}

Deg^°

Rad

Math Response

Answer to #6: $$\frac{y^5}{y^2}$$

Type answer here

Correct Answer: y^3

Deg^°

Rad

Poll

How is your understanding of this topic?