**Learning Targets:**

- I can use the exponent rules for bases other than 10.

**Related Pages**

Illustrative Math

Grade 8

Let’s explore exponent patterns with bases other than 10.

Illustrative Math Unit 8.7, Lesson 6 (printable worksheets)

The following examples show how to use the exponent rules for bases other than 10.

Is each statement true or false? Be prepared to explain your reasoning.

Complete the table to show what it means to have an exponent of zero or a negative exponent.

- As you move toward the left, each number is being multiplied by 2. What is the multiplier as you move toward the right?
- Use the patterns you found in the table to write 2
^{-6}as a fraction. - Write 1/32 as a power of 2 with a single exponent.
- What is the value of 2
^{0}? - From the work you have done with negative exponents, how would you write 5
^{-3}as a fraction? - How would you write 3
^{-4}as a fraction?

- Find an expression equivalent to (2/3)
^{-3}but with positive exponents. - Find an expression equivalent to (4/5)
^{-8}but with positive exponents. - What patterns do you notice when you start with a fraction to a negative power and rewrite it so that it has only positive powers? Show or explain your reasoning.

Lin, Noah, Diego, and Elena decide to test each other’s knowledge of exponents with bases other than 10. They each chose an expression to start with and then came up with a new list of expressions; some of which are equivalent to the original and some of which are not.

Choose 2 lists to analyze. For each list of expressions you choose to analyze, decide which expressions are not equivalent to the original. Be prepared to explain your reasoning.

- Lin’s original expression is 5
^{9}and her list is: - Noah’s original expression is 3
^{10}and his list is: - Diego’s original expression is x
^{4}and his list is: - Elena’s original expression is 8
^{0}and her list is:

- Priya says “I can figure out 5
^{0}by looking at other powers of 5. 5^{3}is 125, 5^{2}is 25, then 5^{1}is 5.”

a. What pattern do you notice?

b. If this pattern continues, what should be the value of 5^{0}? Explain how you know.

c. If this pattern continues, what should be the value of 5^{-1}? Explain how you know. - Select all the expressions that are equivalent to 4
^{-3}. - Write each expression using a single exponent.
- Andre sets up a rain gauge to measure rainfall in his back yard. On Tuesday, it rains off and on all day.

- He starts at 10 a.m. with an empty gauge when it starts to rain.
- Two hours later, he checks, and the gauge has 2 cm of water in it.
- It starts raining even harder, and at 4 p.m., the rain stops, so Andre checks the rain gauge and finds it has 10 cm of water in it.
- While checking it, he accidentally knocks the rain gauge over and spills most of the water, leaving only 3 cm of water in the rain gauge.
- When he checks for the last time at 5 p.m., there is no change.

a. Which of the two graphs could represent Andre’s story? Explain your reasoning.

b. Label the axes of the correct graph with appropriate units.

c. Use the graph to determine how much total rain fell on Tuesday.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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