Lesson 15
Equivalent Exponential Expressions
Let's investigate expressions with variables and exponents.
15.1: Up or Down?
Find the values of \(3^x\) and \(\left(\frac13\right)^x\) for different values of \(x\). What patterns do you notice?
| \(x\) | \(3^x\) | \(\left(\frac13\right)^x\) |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 |
15.2: What's the Value?
Evaluate each expression for the given value of \(x\).
-
\(3x^2\) when \(x\) is 10
-
\(3x^2\) when \(x\) is \(\frac19\)
-
\(\frac{x^3}{4}\) when \(x\) is 4
-
\(\frac{x^3}{4}\) when \(x\) is \(\frac12\)
-
\(9+x^7\) when \(x\) is 1
-
\(9+x^7\) when \(x\) is \(\frac12\)
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.)
- \(64=x^2\)
- \(64=x^3\)
- \(2^x=32\)
- \(x=\left( \frac25 \right)^3\)
- \(\frac{16}{9}=x^2\)
- \(2\boldcdot 2^5=2^x\)
- \(2x=2^4\)
- \(4^3=8^x\)
List:
\(\frac{8}{125}\)
\(\frac{6}{15}\)
\(\frac{5}{8}\)
\(\frac89\)
1
\(\frac43\)
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 \(x\) as a variable. We evaluated these expressions for different values of \(x\).
- To evaluate the expression \(2x^3\) when \(x\) is 5, we replace the letter \(x\) with 5 to get \(2 \boldcdot 5^3\). This is equal to \(2 \boldcdot 125\) or just 250. So the value of \(2x^3\) is 250 when \(x\) is 5.
- To evaluate \(\frac{x^2}{8}\) when \(x\) is 4, we replace the letter \(x\) with 4 to get \(\frac{4^2}{8} = \frac{16}{8}\), which equals 2. So \(\frac{x^2}{8}\) has a value of 2 when \(x\) is 4.
We also saw equations with the variable \(x\) and had to decide what value of \(x\) would make the equation true.
- Suppose we have an equation \(10 \boldcdot 3^x = 90\) and a list of possible solutions: \({1, 2, 3, 9, 11}\). The only value of \(x\) that makes the equation true is 2 because \( 10 \boldcdot 3^2 = 10 \boldcdot 3 \boldcdot 3\), which equals 90. So 2 is the solution to the equation.