6.1 – Properties of Exponents Today’s learning goal: students will be able to use zero and negative exponents Use properties of exponents Solve real –life problems involving exponents.

Core Concept Example: 4 −2 = 1 4 2 Example: 𝑎 −𝑛 = 1 𝑎 𝑛 , where a≠0 Zero Exponent For any nonzero number a, a0 = 1. The power of 00 is undefined. Example: 40 =1 Example: a0 = 1, where a≠0 Negative Exponent For any integer n and any nonzero number a, a-n is the reciprocal of an. Example: 4 −2 = 1 4 2 Example: 𝑎 −𝑛 = 1 𝑎 𝑛 , where a≠0

Example 1 - Example 2- c. 4𝑥 0 𝑦 −3 Evaluate each expression a. 6.7 0 = b. (−2) −4 = 1 (−2) 4 = 1 −2∗−2∗−2∗−2 1 Example 2- = 1 16 Simplify the expression c. 4𝑥 0 𝑦 −3 = 4𝑥 0 𝑦 3 =4 𝑦 3

You Try- Evaluate each expression 1. (−9) 0 = b. 3 −3 = 1 3 3 = 1 3∗3∗3 = 1 27 1 Simplify the expression using only positive exponents c. −5 0 2 −2 d. 3 −2 𝑥 −5 𝑦 0 = 1 3 2 𝑥 5 = 1 9𝑥 5 = −1∗2 2 =−4

Core Concept – Product of Powers Property To multiply powers with the same base, add their exponents. 4 6 ∗ 4 3 = 4 9 To prove this, lets look at the expanded form of the equation: 4 6 means 4∗4∗4∗4∗4∗4 4 3 means 4∗4∗4 So, 4 6 ∗ 4 3 means 4∗4∗4∗4∗4∗4∗4∗4∗4 which is equivalent to 4 9

Example 3 – multiplying with exponents = 3∗3∗3∗3∗3∗3∗3∗3= 3 8 b. 𝑥 3 ∗ 𝑥 5 = 𝑥 8 c. 𝑥 𝑚 ∗ 𝑥 𝑛 = 𝑥 𝑚+𝑛 d. 2 𝑎 ∗ 2 𝑏 = 2 𝑎+𝑏

Core Concept – Quotient of Powers Property To divide powers with the same base, subtract their exponents 4 6 4 3 = 4 6−3 = 4 3 To prove this, lets look at the expanded form of the equation: 4∗4∗4∗4∗4∗4 4∗4∗4 = 4 3

Example 4 – dividing with exponents Evaluate each expression a. (−4) 2 (−4) 7 b. 5 9 5 6 = = 1 (−4) 5 =− 1 −4∗−4∗−4∗−4−∗4 =(−4) 2−7 = (−4) −5 =− 1 1024 5 9−6 = 5 3 =125 c. 3 3 ∗ 5 3 3∗5 2 = 3∗3∗3∗5∗5∗5 3∗5∗5 =3∗3∗5=45

Core Concept – power of a power property To find a power of a power, multiply the exponents ( 4 6 ) 3 = 4 18 To prove this, lets look at the expanded form of the equation: ( 4 6 ) 3 means 4 6 ∗ 4 6 ∗ 4 6 Which would be 4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗4∗ 4∗4∗4∗4∗4 So, ( 4 6 ) 3 means 4 18

Example 4 – Power to a Power Evaluate each expression a. ( 𝑧 4 ) −3 b. ( 6 −2 ) −1 = = 1 𝑧 12 =𝑧 4∗−3 = 𝑧 −12 6 −2∗−1 = 6 2 =36 c. ( 𝑤 12 ) 5 = 𝑤 60

Core Concept – Power of a product To find a power of a product, find the power of each factor and multiply (3∗2) 5 = 3 5 ∗ 2 5 or (𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 Example 5 Evaluate each expression a. (−1.5𝑦) 2 b. ( 𝑎 −10 ) 3 = =(−1.5) 2 ∗ 𝑦 2 = 2.25𝑦 2 𝑎 3 (−10) 3 =− 𝑎 3 1000 c. ( 3𝑑 2 ) 4 = 3 4 𝑑 4 2 4 = 81 𝑑 4 16

Example 5 – Solving a real-life problem A jellyfish emits about 1.25 x 10 8 particles of light, or photons, in 6.25 x 10 −4 second. How many photons does the jellyfish emit each second? Write you answer in scientific notation and in standard form. a. 1.25 𝑥 10 8 6.25 𝑥 10 −4 = 1.25 6.25 𝑥 10 8 10 −4 =0.2 𝑥 10 12 =2 𝑥 10 11 The jellyfish emits 200,000,000,000 photons per second.

You try! It takes the Sun about 2.3 x 10 8 years to orbit the center of the Milky Way. It takes Pluto about 2.5 x 10 2 years to orbit the Sun. How many times does Pluto orbit the Sun while the Sun completes one orbit around the center of the Milky Way? Write you answer in scientific notation. 2.3 𝑥 10 8 2.5 𝑥 10 2 = 2.3 2.5 𝑥 10 8 10 2 =0.92 𝑥 10 6 =9.2 𝑥 10 5