1. The document provides examples for evaluating expressions with scientific notation, square roots, and cube roots. It includes step-by-step workings for finding square roots of perfect squares and cube roots of perfect cubes. 2. Key concepts explained are square roots, perfect squares, radical signs, cube roots, and perfect cubes. Examples are provided to find square roots of numbers less than 100 and cube roots of small numbers. 3. Knowing how to find the exact square roots of perfect squares helps estimate the square roots of nearby non-perfect squares. Being able to write the perfect square root without the radical provides understanding and allows for computations.

1. Course 3, Lesson 1-8
Evaluate each expression. Express the result in scientific
notation.
1. (3.2 × 106)(2.8 × 102)
2.
3. (1.98 × 104) + (3.06 × 106)
4. (2.99 × 1012) – (7.28 × 109)
5. The empty weight of one airplane is 8.35 × 104 pounds, and
the empty weight of a second airplane is 3.07 × 105 pounds.
What is the difference in the empty weights of these two
airplanes?
7
5
4.4 10
1.6 10



2. Course 3, Lesson 1-8
ANSWERS
1. 8.96 × 108
2. 2.75 × 102
3. 3.0798 × 106
4. 2.98272 × 1012
5. 2.235 × 105 pounds

3. WHY is it helpful to write numbers in
different ways?
The Number System
Course 3, Lesson 1-8

4. Course 3, Lesson 1-8 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of
Chief State School Officers. All rights reserved.
The Number System
• 8.EE.2
Use square root and cube root symbols to represent solutions to
equations of the form x2 = p and x3 = p, where p is a positive rational
number. Evaluate square roots of small perfect squares and cube
roots of small perfect cubes. Know that 2 is irrational.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.

5. To
• find the square roots of perfect squares,
• solve equations with square and cube
roots
Course 3, Lesson 1-8
The Number System

6. • square root
• perfect square
• radical sign
• cube root
• perfect cube
Course 3, Lesson 1-8
The Number System

7. Need Another Example?
Step-by-Step Example
1
1. Find the square root.
√64
Find the positive square
root of 64; 82 = 64.
√64 = 8

8. Answer
Need Another Example?
15
Find √225.

9. Need Another Example?
Step-by-Step Example
1
2.
Find both square roots of 1.21;
1.12 = 1.21.
± √1.21 = ± 1.1
Find the square root.
± √1.21

10. Answer
Need Another Example?
±1.2
Find ±√1.44.

11. 1
Need Another Example?
Step-by-Step Example
3. Find the square root.
Find the negative square root of

12. Answer
Need Another Example?

13. 1
Need Another Example?
Step-by-Step Example
4.
There is no real square root because no number
times itself is equal to –16.
Find the square root.
√ –16

14. Answer
Need Another Example?
no real square root
Find √ –81.

15. 1
Need Another Example?
2
3
Step-by-Step Example
5. Solve t2 = 169. Check your solution(s).
t2 = 169 Write the equation.
t = 13 and –13
Definition of square root
Check 13 · 13 = 169 and (–13)(–13) = 169
t = ±√169

16. Answer
Need Another Example?
x2 = 225
±15

17. Course 3, Lesson 1-8
The Number System
Words A square root of a number is one of its two equal factors.
Symbols
Example
2If = ,then is a square root of .x y x y
25 =25 so 5 is a square root of 25.

18. 1
Need Another Example?
Step-by-Step Example
6. Find the cube root.
√125
53 = 5 · 5 · 5 or 125
3
√125 = 5
3

19. Answer
Need Another Example?
3
Find √27.
3

20. 1
Need Another Example?
Step-by-Step Example
7.
(–3)3 = (–3) · (–3) · (–3) or –27
Find the cube root.
√ –27
3
√–27 = –3
3

21. Answer
Need Another Example?
–10
Find √ –1,000.
3

22. 1
Need Another Example?
2
3
4
5
Step-by-Step Example
8. Dylan has a planter in the shape of a cube that
holds 8 cubic feet of potting soil. Solve the equation
8 = s3 to find the side length s of the container.
Write the equation.8 = s3
Take the cube root of each side.
So, each side of the container is 2 feet.
Definition of cube root2 = s
Check (2)3 = 8
√8 = s
3

23. Answer
Need Another Example?
A desktop organizer that is shaped like a cube
has a volume of 125 cubic inches. What is
the length of one side of the organizer?
5 in.

24. Course 3, Lesson 1-8
The Number System
Words A cube root of a number is one of its three equal factors.
Symbols 3If = ,then is the cube root of .x y x y

25. How did what you learned
today help you answer the
WHY is it helpful to write numbers in
different ways?
Course 3, Lesson 1-8
The Number System

26. How did what you learned
today help you answer the
WHY is it helpful to write numbers in
different ways?
Course 3, Lesson 1-8
The Number System
Sample answers:
• You can understand the value of a perfect square root
better when it is written without the radical sign.
• You can complete a computation using numbers when
the perfect square root is written without the radical
sign.

27. The next lesson is about
estimating square roots of
non-perfect squares. Write
how finding the square root
of a perfect square might
help you estimate the
square root of a number
that is not a perfect square.
Ratios and Proportional RelationshipsThe Number System
Course 3, Lesson 1-8