Download A1 CH10 Cubed Roots (1)

Cube Roots
Name:________________________________
Recently, you worked with squares and square roots. You learned that these are inverse operations
because they “undo” each other. The problem 16 means “What number squared would equal 16?”
You found that there are TWO answers, 4 and -4 because both 4 ∙ 4 = 16 , and −4 ∙ −4 = 16.
It was helpful to memorize some Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc.
Now let’s find numbers which are perfect cubes!
DEFINTION: A number, p, is a perfect cube if x3 = p.
The x in this case is called the cube root.
What does that mean? Well, let’s use the chart below to help explore this concept.
1.
Fill in the following table, you may use your calculator.
Cube Root (x)
x•x•x (or x3)
Perfect Cube (p)
1
1•1•1
1
2
2•2•2
3
3•3•3
4
5
6
7
8
9
10
2.
Now use the table values to fill in the blanks below to make true statements.
a. The cube root of 27 is _______ because _____•_____•_____ = 27.
b. Because _____•_____•_____ = 729, the cube root of 729 is _______.
c. 125 is a perfect cube because x3 equals 125 for x = _______, the cube root of 125.
d. 1,000,000 is a perfect cube. What is the cube root of 1,000,000? _______ (Remember, to find
the cube root, you must find the value of x such that x3 = 1,000,000.)
e. The cube root of 216 is _______ because _____•_____•_____ = 216.
f. Because _____•_____•_____ = 512, the cube root of 512 is _______.
g. 64 is a perfect cube because x3 equals 64 for x = _______, the cube root of 64.
3.
!
The expression:
𝑥 is called the cube root of a number x. It asks the question, “What number
cubed would equal x?” So, from the table on the previous page, you found that the cube root of
!
8 = 2. This would be expressed as:
8 = 2.
Now, let’s check
!
8 on the calculator.
Press the MATH button
Select option 4
Type 8 (enter)
You should get the answer 2. Remember, taking the cube root of a number is the inverse
operation of cubing, so it checks because 2! = 8.
Without knowing it, you just solved the equation: 𝑥 ! = 8. In order to get the ‘x’ by itself, you
need to “undo” the cubing by taking the cubed root of both sides. The notation looks like this:
𝑥 ! = 8
!
!
𝑥 ! = 8
𝑥=2
4.
Find the following answers using the cubed root on your calculator.
a.
5.
!
50653
b.
!
6967.871
c.
!
−4096
Going back to your previous lessons with Square Roots, solve: 𝒙𝟐 = 𝟒. (Hint: Take the square
root of both sides.)
How many answers are there for ‘x’ that make the equation true? EXPLAIN WHY.
When you solved 𝒙𝟑 = 𝟖, did you get both a positive and a negative answer for ‘x’?
EXPLAIN WHY OR WHY NOT.
6.
Let’s think about when it IS possible to have a negative cube root.
!
For example, -1•-1•-1 = -1, so the cube root of -1 is -1. This can be written as: −1 = −1.
!
Similarly, -2•-2•-2 = -8, so the cube root of -8 is -2. This is written as: −8 = −2.
a. Because -3 • -3 • -3 = -27, the cube root of -27 = _______.
b. The cube root of -125 is _______ because _____•_____•_____ equals -125.
Therefore, it is possible to get a negative cube root for an answer.
7.
8.
Solve the following equations while showing work. How do you get the variable by itself? What
you do to one side of the equation, you must do to the other. Use your calculator.
a. 𝑥 ! = 64
b.
𝑐 ! = 1728
d. 𝑤 ! = −343
e. 𝑥 ! = −1000
𝑑 ! = −1331
c.
f. 𝑥 ! = 250.047
Sometimes, you’ll have equations where the variable is under the radical sign. In the examples
below, the radical is a cube root (not a square root). To “undo” the cube root, simply cube both
!
sides of the equation. In other words, If you cube 5𝑥 + 3 you get 5x + 3. Solve the following
radical equations, showing complete work.
a.
d.
!
!
𝑥 = 6
b.
!
𝑛 = −9
5𝑥 + 3 = 2
e.
!
2𝑥 − 5 = 3
c.
!
2𝑚 = 4
CHALLENGE:
9.
Suppose the radical sign was not by itself on one side like the examples above. If the radical sign
isn’t isolated, then you need to move everything else to one side, then cube both sides. Solve the
following equations by getting the radical sign by itself first.
a.
!
𝑥+2=6
b.
!
𝑥 − 5 − 4 = −2
!
e.
!
5𝑐 − 6 + 3 = 4
d. −6 + 𝑛 + 8 = −3
c.
!
4𝑦 + 3 = 5