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Chapter 4 Roots &
Powers
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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Table of Perfect Squares, Square Roots, and Cube Roots
Root
1
Perfect Square
1x1=1
Perfect Cube
1x1x1=1
2
4
8
3
4
5
6
7
8
9
10
Vocabulary List
For each word, provide a definition in your own words and an example.

Radicand

Radical
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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
Index

Whole Numbers

Exact Value

Approximate Value

Natural Numbers
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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
Rational Numbers

Irrational Numbers

Integers
Mixed radical
Entire radical
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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Section 4.1: Estimating Roots
Estimating Roots:
1) Write the two consecutive (numbers that are in a row like 1 and 2, 5 and 6, etc) perfect squares
(or perfect cubes, etc) closest to the radicand. One perfect square (cube, etc) is larger, one is
smaller.
2) Complete a table by estimating the square root (cube root, etc) and finding the square (cube,
etc) of this value until your answer is within 1 decimal place (or whatever is asked for) of the
radicand.
a. If the value of the square is too large, reduce the estimated root.
b. If the value of the square is too small, increase the estimated root.
c. Use your best judgment to figure out how much to increase or decrease by each time. IT
IS GUESS AND CHECK!
3) Justify your answer.
Example 1: Find the estimated value of
20 within 1 decimal place of 20.
1) Write two consecutive perfect squares closest to 20.
______________ _________________
2) Fill in the table until the estimate of the square is within 1 decimal place of 20.
Estimated
Value of its square
20
20 = _____________
3) Justify your answer. _____________________________
What does it mean to justify your answer?
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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Example 2: Find the estimated value of
3
20 within 1 decimal place of 20.
1) Write two consecutive perfect cubes closest to 20. (Hint: Check your reference list).
______________
___________________
2) Fill in the table until the estimate of the cube root is within 1 decimal place of 20.
Estimated
3
3
Value of its cube
20
20 = ______________
3) Justify your answer. _____________________________
Question 3: Find the estimated fourth root of 20 within 1 decimal place of 20.
1) Write two consecutive perfect fourth powers closest to 20.
_____________________
__________________________
2) Fill in the table until the estimate of the fourth root is within 1 decimal place of 20.
Estimated
4
20
Foundations and Pre-Cal Math 10
Estimates fourth power
Chapter 4: Roots and Powers
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4
20 = ________________
3) Justify your answer. _____________________________
Question 4: Complete the table below:
Radical
Value
Is the Value Exact or Approximate?
16
4
Exact
27
5.1962
Approximate
16
81
4
9 or 0.4̄
Exact
0.64
3
16
3
27
3
16
81
3
0.64
3
0.64
4
16
4
27
4
4
16
81
0.64
What does it mean to write an equivalent form as a square root?
Complete in Vocabulary List: Radicand, Radical, Index, Whole Number, Exact Value, Approximate
Value
HW: Complete Page 206 #2 - #6
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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4.2 Irrational Numbers
What does a decimal number that terminates look like? ____________________
What does a decimal number that repeats look like? _______________________
Steps to determine if a number is irrational or rational:
1) Use your calculator to find the value (approximate or exact) of the number.
2) Determine if it is a repeating or terminating decimal (rational) or not (irrational).
Example 1: Classifying Numbers
Tell whether each number is rational or irrational. Explain how you know!
a) 
3
5
b)
d)
8
27
e) 3
3
c)
14
1
9
f) 0
Why can’t an irrational number have a 0 in the denominator?
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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Together, the rational and irrational numbers form the set of real numbers!
Steps to order numbers:
1) First put the numbers all in decimal form.
2) Order the numbers (remember to think about negatives!)
Example 2: Ordering Irrational Numbers on a Number Line
Use a number line to order these numbers from least to greatest. Also decide if they are rational or
irrational (use R and I), exact or approximate (use E and A).
Vocabulary List: Natural Numbers, Rational Number, Irrational Numbers, Integers
HW: Pg. 211 #3, 4, 5, 7, 8, 10, 11, 12, 14, 15
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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4.3 Mixed and Entire Radicals (Day 1)
To multiply radicals, you can either multiply the radicands beneath the radical or you can
separate these factors into 2 or more radicands.
Example:
Steps to Simplifying Radicals Using (also referred to as changing an entire radical into a mixed
radical)
1) Use prime factorization (from chapter 3) to find the factors of the radical. Then find any
repeating factors.
a. For square roots, you need 2 of the same factor. (Factor out a perfect square)
b. For cube roots, you need 3 of the same factor. (Factor out a perfect cube)
c. Etc
2) “Factor” out the perfect square by moving the 2 identical factors from underneath the radical
sign to 1 factor outside the radical sign. (similar for cubes, etc)
3) Do this for ALL factors that you can.
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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Can a radical always be simplified? Why or why not?
HW: Page 218 # 4, 10, 11, 14, 15, 16, 17
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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4.3 Mixed and Entire Radicals (Day 2)
Steps for writing mixed radicals as entire radicals.
1) Use the Multiplication Property of Radicals to change the number in front of the radical (called
the coefficient) into part of the radicand (number inside the radical symbol).
a. To do this, square the number in front if it is a square root. (Cube the number in front if
it is a cube root, etc).
Vocabulary List: Mixed Radical, Entire Radical
HW: Page 218 # 12acegi, 18ac, 8, 22a (no decimals!), 24
Assessment: Quiz (4.1 to 4.3)
Foundations and Pre-Cal Math 10
Chapter 4: Roots and Powers
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