Simplifying Square Roots

Show your work for each problem.

1.
What is ?
1. a)18
2. b)6
3. c)9
4. d)12
Answer: 6
because . 36 is a perfect square.
2.
Which is the largest perfect square factor of 50?
1. a)50
2. b)25
3. c)10
4. d)5
Answer: 25
. Since is a perfect square and 25 divides 50 evenly, 25 is the largest perfect square factor.
3.
Simplify . Enter the coefficient (the number in front of the square root).
Answer: 2
. The coefficient is 2.
4.
Simplify . What number stays under the radical?
Answer: 2
. The number 2 stays under the radical.
5.
Simplify step by step.
Answer: 2sqrt(3)
1. What is the largest perfect square that divides 12? 4
2. Write 12 as a product: 3
3. What is ? 2
4. So . What is the coefficient? 2
6.
What is in simplified form?
1. a)
2. b)
3. c)
4. d)
Answer:
. The other options are not fully simplified.
7.
Which square root is already in simplest form?
1. a)
2. b)
3. c)
4. d)
Answer:
has no repeated prime factors, so cannot be simplified. The others can: , , .
8.
Simplify . Enter the coefficient.
Answer: 5
. The coefficient is 5.
9.
Simplify . Enter the coefficient.
Answer: 7
. The coefficient is 7.
10.
Simplify using prime factorization.
Answer: 5sqrt(3)
1. Find the prime factorization of 75: 25
2. 5
3. So . What is ? 5
4. Therefore 5
11.
Simplify step by step.
Answer: 6sqrt(5)
1. What is the prime factorization of 180? 5
2. Which factors appear in pairs? (Answer: 2 and ?) 3
3. What is ? 2
4. What is ? 3
5. Multiply: 6
6. So 6
12.
What is in simplified form?
1. a)
2. b)
3. c)
4. d)
Answer:
. The other options are not fully simplified.
13.
Simplify . Enter the coefficient.
Answer: 7
. The coefficient is 7.
14.
Simplify . Enter the coefficient.
Answer: 14
. The coefficient is 14.
15.
Find the diagonal of a square with side length 8. Give your answer in simplified radical form.
Answer: 8sqrt(2)
1. Using the Pythagorean theorem, the diagonal satisfies . What is ? 64
2. What is ? 128
3. So . What is the largest perfect square factor of 128? 64
4. What is ? 8
5. So the diagonal is 8
16.
Simplify step by step.
Answer: 12sqrt(5)
1. Find the prime factorization: 5
2. What is ? 4
3. What is ? 3
4. Multiply: 12
5. So 12