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Math 97 Supplement 2

LEARNING OBJECTIVES

1. Solve equations by using inverse operations, including squares, square roots, cubes,

and cube roots.

The Definition of Inverse Operations

A pair of inverse operations is defined as two operations that will be performed on a number or variable, that always results in the original number or variable. Another way to think of this is

For example, addition and subtraction are

inverse operations since we can say 22xx
. If we start with x, then add 2 and subtract 2, we are left with the original starting variable x. There are several inverse operations you should be familiar with: addition and subtraction, multiplication and division, squares and square roots (for positive numbers), as well as cubes and cube roots. The following examples summarize how to undo these operations using their inverses. Using Inverse Operations with the 4 Basic Operations

Addition Subtraction Multiplication Division

Solve:

23x
x has 2 added to it, so we subtract 2 from both sides. 23
22
x

Solution:

1x

Solve:

23x
x has 2 subtracted from it, so we add 2 to both sides. 23
22
x

Solution:

5x

Solve:

28x
x has 2 multiplied to it, so we divide 2 from both sides. 28
22
x

Solution:

4x

Solve:

82
x x is divided by 2, so we multiply by 2 on both sides.

2 8 22

x

Solution:

16x 2

Using Inverse Operations with Powers and Roots

Square Root Square Cube Root Cube

Solve:

4x x is being square rooted, so we square both sides. 224x

Solution:

16x

Solve:

24x
x is being squared, so we square root both sides. (using root) 24x

Solution:

2x or 2x

Solve:

32x
x is being cube- rooted, so we cube both sides. 3332x

Solution:

8x

Solve:

38x
x is being cubed, so we cube root both sides. 3338x

Solution:

2x Note that undoing the square with a square root required both a positive and a negative in front of the root. That is because when we square a positive or a negative number we get a positive. x in 24x
should be a +2 or a -2 since both of these make the original equation true: 224
and 224
. So, we include both +2 and -2 as an answer.

Also note that w

with the cube root since only a positive cubed would give us a positive. In other words, 328
, but 328
, so we just need the positive cube root.

Example 1

Solve the following:

a. 9x b. 29x
c. 39x

Solution:

a.

29 81x

b. Ԝ 93x
c. 39x
3

Consider the following equation:

3 2 12x

There are two ways to solve this problem, and both of them require eliminating the parentheses. One method is to use the distributive property, and the other is to use inverse operations. The chart below shows a comparison of these techniques. Using Distributive Property Using Inverse Operations Only Solve

3 2 12x

Distribute the 3 through the parentheses

3 6 12x

Now use inverse operations by adding 6 to get

the 3x isolated on the left side

3 6 12

6 6

3 18 x x

Isolate the x by dividing both sides by 3

3 18 33
x

Solution:

6x Solve

3 2 12x

Divide both sides by 3 to isolate the parentheses

3212
33
24
x x

Now we can remove the parentheses since it is

alone on the left, then add 2 on each side 24

2 2

x

Solution:

6x

Now consider this similar problem: Solve

23 2 12x

This one cannot be solved by distributing the 3, we have to use inverse operations on this one.

Example 2

Solve

23 2 12x

2 2

3212 33

2 4 2 4 2 2 xDivide both sides by 3 x Undo the square with a square root x Add 2 on both sides, simplify the root x Simplify the + and the r r

4 or 0 x x Final answers!

4 , think about the following computations: 222
2

3 4 3 4

7 9 16

49 25
22
22
2

2 3 4 2 3 2 4

2 7 6 8

2 49 14

98 196

The bottom line is obviously false, and so are all of the previous lines. The same is true for roots:

9 16 9 16

25 3 4

57

4 9 16 4 9 4 16

4 25 36 64

4 5 100

20 10 CANNOT distribute a number through a power or a root, and you cannot distribute a power or a root to each term inside. This means we will only be using inverse operations to solve equations with powers or roots for now.

Example 3

Solve

32 7 1x

Solution:

3 3 3 2 7 1

7 7

2 6 3 x

Subtract 7 on both sides to isolate the root term

x Divide both sides by 2 to isolate the root x Cube both sides to undo the cube-root x 33
27

Simplify

x Final answer!

Try this! Solve:

2 5 3x

Answer:

7x 5

Example 4

Solve

33 2 1 192x

Solution:

3 3 3

3 2 1192 33

2 1 = 64

2 1 64

2 1 4 xDivide by 3 on both sides to isolate the square x Take the cube root of both sides x Simplify the root if possible x 2 3

3 2

Add 1 on both sides to isolate the x-term

x Divide by 2 on both sides to isolate the x x Final Answer!

Example 5 and 6 to nice whole

numbers.

Example 5

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