Understand and Represent Exponents

Key Concepts

After this lesson, students will be able to:

• Understand and Represent Exponents.
• Identify the base and exponent in a power.
• Write powers for repeated multiplication
• Write powers in expand and word form
• Evaluate Exponents
• Evaluate Expressions with Exponents.

Essential Question  

How can you write and evaluate numbers with exponents? 

EXAMPLE 1: 

 2 x 2 x 2 represents the number of cells after 1 hour if there is 1 cell at the start.  

How can you write this expression using exponents? How many cells will there be after 1 hour? 

Reasoning  

Repeated multiplication can be represented in more than one way. 

You can use an exponent to write a repeated multiplication of a number.  

A number that can be written using exponents is called a power. 

You can use repeated multiplication to evaluate, or find the value of a power. 

Complete the table below: 

Try It!  

There are 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 bacteria cells after 3 hours.  Write the repeated multiplication as a power, and then evaluate. 

Convince Me!  

Why can you represent the number of cells after two hours as the power 2⁶ ?

Yes, First hour = 2 x 2 x 2 = 2³ = 8 

Second hour = 2 times 2 x 2 x 2= (2 x 2 x 2) x (2 x 2 x 2) = 2⁶ = 64 

  Third hour = 3 times 2 x 2 x 2 = (2 x 2 x 2)   x  (2 x 2 x 2 ) x ( 2 x 2 x 2 )= 2⁹ =512 

Evaluate Exponents  

EXAMPLE 2: 

1. How can you evaluate 2°?  

The base is 2. The exponent is 0. Make a table and look for a pattern. 

Each value equals the previous value multiplied by 2.  

1 x 2 = 2, so the value of 2° is 1.  

Generalize,  

Any non-zero number raised to an exponent of zero has a value of 1. 

2. How can you evaluate 1.2⁴ ?

Solution: 

The base is 1.2. The exponent is 4. 

1.2⁴ = 1.2 x 1.2 x 1.2 x 1.2 

         = 1.44 x 1.2 x 1.2  (∵Multiply by the first two factors) 

         = 1.728 x 1.2          (∵Multiply by the third factor) 
∴1.2⁴ =2.0736             (∵Multiply by the fourth factor) 

1.2⁴ = 2.0736 

EXAMPLE 3: 

Julia calculated the foil as 1.9 x 10⁵units thick. Thom calculated the foil as 183,000 units thick. Which calculation represents the greater thickness for the foil?  

Evaluate the expression: 1.9 x10⁵

Solution: 

10⁵= 10 x 10 x 10 x 10 x 10 = 100000 

Multiply by the decimal: 1.9 X 100,000 = 190,000 

Compare the numbers. 

190,000 > 183,000 Julia’s calculation represents the greater thickness for the foil. 

Try It!  

Rafael calculated the foil as 1.8 x10⁵ units thick. Evaluate Rafael’s expression. 

Solution: 

1.8 x10⁵ =1.8 x 100000 =180,000. 

Practice & Problem Solving: 

Write the exponent for each expression 

10. A company rents two storage units. Both units are cube-shaped. What is the difference in the volume of the two storage units? Note that the volume of a cube is s3, where s is the side length. Explain. 

Solution: 

Given that, 

Two storage units which are cube shaped with side lengths of 8 ft and 6.5 ft, respectively. 

Volume of the first storage unit = s³

                                                   = (8)³

                                                 =8 x 8 x 8 

                                                 =512 square fts 

Volume of the second storage unit = s³
= (6.5)³

                                                              =6.5 x 6.5 x 6.5 

                                                              =274.625 square fts 

Difference of two storage units in volume 

= 512 square fts – 274.625 square fts 

=237.375 square fts. 

∴Difference of two storage units in volume = 237.375 square fts.  

11. Malik read that the land area of Alaska is about 5.7 x 10⁵ square miles. About how many square miles is the land area of Alaska?  

Solution: 

Given that, 

Land area of Alaska = 5.7 x 10⁵

square miles 

5.7 x 10⁵

= 5.7 x 10 x 10 x 10 x 10 x 10 

                 = 5.7 x 100,000  

                 =570,000 

∴The land area of Alaska = 570,000 square miles 

Higher Order Thinking  

12. Zach invested $50 and tripled his money in two years. Kayla also invested $50, and after two years the amount was equal to 50 to the third power. Who had more money after two years? Explain. 

Solution: 

Given that, 

Money invested by Zach =  $50  

Zach tripled his money in two years  

Then,  

Zach money after two years = 3 x 50 

                                                       =150  

Zach money after two years= $150 

Money invested by Kayla = $50 

Her amount was equal to 50 to the third power 

Then, 

Kayla money after two years = 50³

                                                      =50 x 50 x 50 

                                                     = $125,000 

                           ∴Kayla has more money after two years 

Check your knowledge:  

Answer the following 

1. Write 81 as the repeated multiplication of 3s. Then write it as power. 

2. Write 125 as the repeated multiplication of 5s. Then write it as a power. 

3. What is 0.75 x 0.75 x 0.75 x 0.75 written as power? 

What is 3/8 x 3/8 x 3/8 written as power? 

Evaluate each power. 

5. (1/6)²

6. 45⁰

7. 0.1⁵

8. 7³ 

Evaluate each expression. 

9. 4.5 x 10³  

10. 0.6 x 10⁶ 

11. 3.4 x 10⁰

Answers: 

1. 81 = 3 x 27  

            =3 x 3 x 9 

            =3 x 3 x 3 x 3 

            = 3⁴

  ∴ 81 = 3⁴

2. 125 = 5 x 25 

              = 5 x 5 x 5 

              = 5³

 ∴ 125 = 5³ 

3. 0.75 x 0.75 x 0.75 x 0.75 = 0.75⁴

4. 3/8 x 3/8 x 3/8 = (3/8)³ 
5. (1/6)² = 1/6 x 1/6 
   1/36

6. 45⁰ = 1 

7. 0.1⁵= 0.1 x 0.1 x 0.1 x 0.1 x 0.1  

        =0.00001 

8. 7³= 7 x 7 x 7 
   = 343 

9. 4.5 x 10³= 4.5 x 10 x 10 x 10  
   =4.5 x 1000 
   =4500 

10. 0.6 x 10⁶ = 0.6 x 10 x 10 x 10 x 10 x 10 x 10  
    =0.6 x 1,000,000 
    =600,000 

11. 3.4 x 10⁰ = 3.4 x 1 
    =3 

Key concept covered 

• Understand and represent exponents. 
• Identify the base and the exponent in a power. 
• Write powers for repeated multiplication. 
• Write powers in expanded and word form. 
• Evaluate exponents. 
• Evaluate expressions with exponents. 

Concept map 

3.1 Understand and Represent Exponents