Lesson 16

Equivalent Exponential Expressions

Let's investigate expressions with variables and exponents.

16.1: 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    

16.2: What's the Value?

Evaluate each expression for the given value of \(x\). 

  1. \(3x^2\) when \(x\) is 10

  2. \(3x^2\) when \(x\) is \(\frac19\)

  3. \(\frac{x^3}{4}\) when \(x\) is 4

  4. \(\frac{x^3}{4}\) when \(x\) is \(\frac12\)

  5. \(9+x^7\) when \(x\) is 1

  6. \(9+x^7\) when \(x\) is \(\frac12\)

16.3: 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.)

  1. \(64=x^2\)
  2. \(64=x^3\)
  3. \(2^x=32\)
  4. \(x=\left( \frac25 \right)^3\)
  5. \(\frac{16}{9}=x^2\)
  6. \(2\boldcdot 2^5=2^x\)
  7. \(2x=2^4\)
  8. \(4^3=8^x\)

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.

A large tetrahedron composed of various colors of smaller tetrahedra.
  1. 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.
  2. How many small tetrahedra are in the bottom layer, touching the table?
  3. 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.
  4. How many small faces would this bigger fractal have? How many small tetrahedra would be in the bottom layer?
  5. 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.

Glossary Entries

  • cubed

    We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).

  • exponent

    In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).

  • squared

    We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).