Lesson 15
Equivalent Exponential Expressions
Let's investigate expressions with variables and exponents.
15.1: Up or Down?
Find the values of and for different values of . What patterns do you notice?
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2 | ||
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15.2: What's the Value?
Evaluate each expression for the given value of .
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when is 10
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when is
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when is 4
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when is
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when is 1
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when is
15.3: Exponent Experimentation
Find a solution to each equation in the list. (Numbers in the list may be a solution to more than one equation, and not all numbers in the list will be used.)
List:
1
2
3
4
5
6
8
This fractal is called a Sierpinski Tetrahedron. A tetrahedron is a polyhedron that has four faces. (The plural of tetrahedron is tetrahedra.)
The small tetrahedra form four medium-sized tetrahedra: blue, red, yellow, and green. The medium-sized tetrahedra form one large tetrahedron.

- How many small faces does this fractal have? Be sure to include faces you can’t see. Try to find a way to figure this out so that you don’t have to count every face.
- How many small tetrahedra are in the bottom layer, touching the table?
- To make an even bigger version of this fractal, you could take four fractals like the one pictured and put them together. Explain where you would attach the fractals to make a bigger tetrahedron.
- How many small faces would this bigger fractal have? How many small tetrahedra would be in the bottom layer?
- What other patterns can you find?
Summary
In this lesson, we saw expressions that used the letter as a variable. We evaluated these expressions for different values of .
- To evaluate the expression when is 5, we replace the letter with 5 to get . This is equal to or just 250. So the value of is 250 when is 5.
- To evaluate when is 4, we replace the letter with 4 to get , which equals 2. So has a value of 2 when is 4.
We also saw equations with the variable and had to decide what value of would make the equation true.
- Suppose we have an equation and a list of possible solutions: . The only value of that makes the equation true is 2 because , which equals 90. So 2 is the solution to the equation.