Algebra 1
Square roots (continued)
Name: __________________________________
Day 1 - Review: What we've learned so far:
Simplifying square roots containing integers:
Ex 1: Simplify:
450
1. Find the prime factors of 450:
450  2  32  52
450  2  32  52
2. Simplify:
 35 2
 15 2
Ex 2: Simplify: 3 72x 3 y
1. Find the prime factors of 72:
72  23  32  2  22  32
2. Simplify:
3 72 x3 y  3 2  22  32  x  x 2  y
 3  2  3  x 2 xy
 18 x 2 xy
Multiplying and simplifying two square roots:
Ex 3: Simplify:
13  52
13  52  13  52
 13 13  2  2
 13  2
 26
Ex 4: Simplify: 2 5x2  6 10 x
2 5 x 2  6 10 x  12 5  x 2 10  x
 12 5  x 2  5  2  x
 12  5  x  2 x
 60 x 2 x
1
Simplifying square roots containing fractions (without rationalizing the denominator):
Ex 5:
16
16 4


9
9 3
Ex. 6:
18
18
29
2  32 3 2




4
4
4
4
16
If possible, reduce first!
Ex. 7:
12
12
4
4 2




75
25
75
25 5
2
Day 2 - Simplifying square roots containing fractions and rationalizing the denominators:
1.
If possible, reduce the fraction by canceling common factors in the numerator and
denominator.
2.
Simplify the numerator and the denominator (remove “buddies” from the square roots).
3.
If a square root remains in the denominator, eliminate it by multiplying both the
numerator and the denominator by the same square root. Eliminating a radical from the
denominator is called rationalizing the denominator.
Examples:
Simplify:
5
,
7
3
,
2
9
,
98
2
,
75
56
12
Ex. 1.
5
5 7
57
35




7
7
7 7
77
Ex. 2.
3
3
2 3 2 3 2




2
2
2 2
22
Ex. 3.
9
33
3
3
2
3 2
3 2 3 2







98
14
277 7 2 7 2 2 7 22 72
Ex. 4.
2
2
2
2
3
6
6
6







75
3  5  5 5 3 5 3 3 5 3  3 5  3 15
Ex. 5.
56

12
4 14
14
14 3
14  3
42





4 3
3
3
3
3
33
3
Day 3 - Adding and subtracting, multiplying by FOILing, rationalizing using conjugates
Adding and subtracting square root expressions:
Adding and subtracting square root expressions is similar to adding and subtracting variable
expressions. We can add or subtract variable terms only if the variables are the same; similarly,
we can add or subtract square root expressions only if the numbers in the square root symbols are
the same:
Can be combined:
Can’t be combined:
3 2 4 2 7 2
5 6  5 6
is similar to
3x  4x  7 x
is similar to
x y  x y
NOTE: The numbers inside the square root symbols do NOT change when we add or
subtract!
Examples:
Ex. 1.
3 2  9 2  12 2
Ex. 2.
3 5  7 5  4 5
For most problems, we will have to simplify the square root expressions before adding or
subtracting:
12  4 3  22  3  4 3
Ex. 3.
Ex. 4.
2 10  7 40  2 10  7 2 2 10
2 34 3
 2 10  7  2 10
6 3
 2 10  14 10
 16 10
Multiplying square root expressions by distributing or FOILing:
Recall that when multiplying square root expressions, you should
a. multiply the numbers outside the square roots together,
b. write the numbers inside the square roots as a product but do not multiply the numbers,
c. factor and simplify the product inside the square root.
These same rules apply when multiplying by distributing or FOILing.
Examples:
Ex. 1.
2 3


6  3 11  2 3  6  2  3 3 11
 2 3  3  2  6 3 11
 2  3 2  6 33
 6 2  6 33
4
Ex. 2.
3
2 3


2  5 3  3 2  2  35 2 3  3 2  5 33
 3  2  15 6  6  5  3
 6  14 6  15
 9  14 6
Rationalizing denominators by multiplying by the conjugate:
Conjugates are the sum and difference of the same two terms. The expressions
7  2 and 7  2 are conjugates. The product of square root conjugates has no square root:
Ex. 1.

7 2


7  2  77  72  27  22
 7  14  14  2
72
5
A denominator that contains a sum or difference that includes square root expressions can be
rationalized by multiplying the numerator and the denominator by the conjugate:
Ex. 2
3
3 10  6


10  6 10  6 10  6

30  3 6
10 10  10 6  10 6  6  6

30  3 6
100  6

30  3 6
94
5
Day 4 - Solving square root equations
Solving an equation with two square roots:
If there is one radical on each side of the equation sign:
1. Square both sides.
2. Solve for the variable.
3. Check for extraneous solutions.
Ex. 1.
3x  4  5 x  6
Solving an equation with one square root:
If there is one radical in the equation, solve the equation as follows:
1. Isolate the radical.
2. Square both sides.
3. Solve for the variable.
4. Check for extraneous solutions.
Ex. 2:
Ex. 4:
x  7  12
x  x6
Ex. 3.
x2 6
Ex. 5:
2 x  20  8
6
Day 1 Homework: Do not use a calculator for these problems!
7
Day 2 Homework: Do not use a calculator for these problems!
8
Day 3 Homework: Do not use a calculator for these problems!
9
Day 4 Homework: Do not use a calculator for these problems!
10
Homework answers:
Day 1:
Day 2:
11
Day 3:
Day 4:
12