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