Lesson 18

Bases and Exponents

  • Let’s rewrite expressions using the property \((x^a)^b = x^{ab}\).

18.1: Math Talk: Different Bases

Decide if each expression is equal to \(9^{16}\).

\((9^{8})^{8}\)

\((9^4)^4\)

\((3^2)^{16}\)

\(3^{32}\)

18.2: What’s the Factor?

  1. Refer to the first table.
    step 0 1 2 3 4 5 6
    value 10 30 90 270                       
    expression \(10\boldcdot3^0\) \(10\boldcdot3^1\) \(10\boldcdot3^2\)                                                
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 4?
      2. from step 3 to step 6?
      3. Conjecture about the factor from step 7 to step 10.
    3. By what factor does the value change . . .
      1. from step 0 to step 5?
      2. from step 1 to step 6?
      3. Conjecture about the factor from step 10 to step 15.
  2. Refer to the second table.
    step 0 1 2 3 4 5 6
    value 3 6 12 24      
    expression \(3\boldcdot2^0\)                                                                                
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 3?
      2. from step 3 to step 5?
      3. Conjecture about the factor from step 10 to step 12.
    3. By what factor does the value change . . .
      1. from step 0 to step 3?
      2. from step 2 to step 5?
      3. Conjecture about the factor from step 10 to step 13.
  3. Refer to the third table.
    step 0 1 2 3 4 5 6
    value 2,048 1,024 512        
    expression                                                                                               
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 3?
      2. from step 3 to step 5?
      3. Conjecture about the factor from step 10 to step 12.
    3. By what factor does the value change . . .
      1. from step 0 to step 3?
      2. from step 2 to step 5?
      3. Conjecture about the factor from step 10 to step 13.

18.3: Rewriting Expressions

  1. For each given expression, decide what to write in the box to create equal expressions.
    given expression equal expression 1 equal expression 2
    \(5\boldcdot10^8\) \(5\boldcdot100^\boxed{\phantom{3}}\) \(5\boldcdot\boxed{\phantom{3}}^2\)
    \(7\boldcdot16^9\) \(7\boldcdot\boxed{\phantom{3}}^{4\boldcdot9}\) \(7\boldcdot4^\boxed{\phantom{3}}\)
    \((0.25)^3\) \((0.5)^\boxed{\phantom{3}}\) \(\boxed{\phantom{3}}^1\)
    \(3\boldcdot(1.2)^6\) \(3\boldcdot1.44^\boxed{\phantom{3}}\) \(3\boldcdot1.728^\boxed{\phantom{3}}\)
    \(6\boldcdot0.09^{10}\) \(6\boldcdot\boxed{\phantom{3}}^5\) \(6\boldcdot0.3^\boxed{\phantom{3}}\)

     

  2. Write at least 3 new expressions that are equal to \(4 \boldcdot 27^6\).

Summary