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:
- 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.
- 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
- 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.
- 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
- 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
- Write 81 as the repeated multiplication of 3s. Then write it as power.
- Write 125 as the repeated multiplication of 5s. Then write it as a power.
- 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.
- (1/6)²
- 45⁰
- 0.1⁵
- 7³
Evaluate each expression.
- 4.5 x 10³
- 0.6 x 10⁶
- 3.4 x 10⁰
Answers:
- 81 = 3 x 27
=3 x 3 x 9
=3 x 3 x 3 x 3
= 3⁴
∴ 81 = 3⁴
- 125 = 5 x 25
= 5 x 5 x 5
= 5³
∴ 125 = 5³
- 0.75 x 0.75 x 0.75 x 0.75 = 0.75⁴
- 3/8 x 3/8 x 3/8 = (3/8)³
- (1/6)² = 1/6 x 1/6
1/36
- 45⁰ = 1
- 0.1⁵= 0.1 x 0.1 x 0.1 x 0.1 x 0.1
=0.00001
- 7³= 7 x 7 x 7
= 343
- 4.5 x 10³= 4.5 x 10 x 10 x 10
=4.5 x 1000
=4500
- 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
- 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