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Lesson 15Equivalent Exponential Expressions

Let's investigate expressions with variables and exponents.

Learning Targets:

  • I can find solutions to equations with exponents in a list of numbers.
  • I can replace a variable with a number in an expression with exponents and operations and use the correct order to evaluate the expression.

15.1 Up or Down?

  1. Find the values of 3^x and \left(\frac13\right)^x for different values of x .
    x   3^x \left(\frac13\right)^x
    1
    2
    3
    4
  1. What patterns do you notice?

15.2 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

15.3 Exponent Experimentation

Find a solution to each equation in the list that follows. (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
  1. \frac{16}{9}=x^2
  2. 2\boldcdot 2^5=2^x
  3. 2x=2^4
  4. 4^3=8^x

List:

\frac{8}{125}

\frac{6}{15}

\frac{5}{8}

\frac89

1

\frac43

2

3

4

5

6

8

Are you ready for more?

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 triangular pyramid is made of smaller traingular pyramids
  1. How many small faces does this fractal have? Be sure to include faces you can’t see as well as those you can. 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?

Lesson 15 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.

Lesson 15 Practice Problems

  1. Evaluate the following expressions if x=3 .

    1. 2^x
    2. x^2
    1. 1^x
    2. x^1
    1. \left(\frac12\right)^x

  2. Evaluate each expression for the given value of x . 

    1. 2 + x^3 , x is 3 
    2. x^2 , x is \frac{1}{2}  
    1. 3x^2 , x is 5 
    2. 100 - x^2 , x is 6

  3. Decide if the expressions have the same value. If not, determine which expression has the larger value.

    1. 2^3 and  3^2
    2. 1^{31} and  31^1
    1. 4^2 and  2^4
    2. \left(\frac12\right)^3 and  \left(\frac13\right)^2

  4. Match each equation to its solution.

    1. 7 + x^2 = 16
    2. 5 - x^2 = 1
    3. 2 \boldcdot 2^3 = 2^x
    4. \frac{3^4}{3^x}=27
    1. x=4
    2. x=1
    3. x=2
    4. x=3

  5. An adult pass at the amusement park costs 1.6 times as much as a child’s pass.

    1. How many dollars does an adult pass cost if a child’s pass costs:

      $5?

      $10?

      w

    2. A child’s pass costs $15. How many dollars does an adult pass cost?

  6. Jada reads 5 pages every 20 minutes. At this rate, how many pages can she read in 1 hour?

    1. Use a double number line to find the answer.
    diagram showing pages read over time in minutes
    1. Use a table to find the answer. 
    pages read time in minutes
    5 20
    1. Explain which strategy you thinks works better in finding the answer.