Square Root = √ Number² = √ Number × Itself
Basics
Square Root = √Number² = √Number × Itself
Square Root of 9 = √9 = √3² = √3 × 3 = 3
Square Root of 144 = √144 = √12² = √12 × 12 = 12
Square Root of 412.09 = √412.09 = √20.3² = √20.3 × 20.3 = 20.3
The root sign √ is also known as the radical sign.
When the root sign √ is immediately adjacent to a number or letter, an invisible multiplication sign is in force.
5√3 = 5 × √3
a√3 = a × √3
is the square root command.
activates 
above 
.
It is possible that questions involving simplifying square roots will be in the calculator-prohibited section of the test. In that case, it is helpful to know your multiplication and division tables.
When simplifying square roots without a calculator, a variation of factoring is used to extract from pairs of numbers under the root sign.
The numbers 2 and 3 are often useful in factoring.
Formulas (NOT on Mathematics Formula Sheet)
Square Root = √Number² = √Number × Itself
√NumberA × √NumberB = √NumberA × NumberB
Question
What is √16 ?
Answer
4
| Input | Display | Comment |
 |
blinker | clears screen |
  16 |
√16 | Square Root |
 |
4 | Answer |
Square Root |
Question
What is √5097.96 ?
Answer
71.4
| Input | Display | Comment |
|
blinker | clears screen |
| 5097.96 |
√5097.96 | Square Root |
|
71.4 | Answer |
Square Root |
Question
Using the calculator, what is the simplified version of √180 ?
Answer
6√5
| Input | Display | Comment |
|
blinker | clears screen |
| 180 |
√180 | Square Root |
|
6√5 | Answer |
Square Root
|
Question
Without using the calculator, what is the simplified version of √180 ?
Answer
6√5
Answer Process
When simplifying square roots without a calculator, a variation of factoring is used to extract from pairs of numbers under the root sign:
Factoring (Variation):
180 ÷ 2 = 90
Multiple Factor
90 ÷ 2 = 45
Multiple Factor
45 ÷ 3 = 15
Multiple Factor
15 ÷ 3 = 5
Multiple Factor
5 ÷ 5 = 1
Multiple Factor
√180 = √2 × 2 × 3 × 3 × 5 = 2 × 3√5 = 6√5
2 is extracted from a pair of 2’s under the root sign;
3 is extracted from a pair of 3’s under the root sign;
an unpaired 5 is “left behind” under the root sign.
Question
Using the calculator, what is the simplified version of
√6 × √12 ?
Answer
6√2
| Input | Display | Comment |
|
blinker | clears screen |
6  × 12 |
√6 *√12 | |
|
6√2 | Answer |
Square Root |
Question
Without using the calculator, what is the simplified
version of √6 × √12 ?
Answer
6√2
Answer Process
√NumberA × √NumberB = √NumberA × NumberB
√6 × √12 = √6×12 = √72 = 6√2
When simplifying square roots without a calculator, a variation of factoring is used to extract from pairs of numbers under the root sign:
Factoring (Variation):
72 ÷ 2 = 36
Multiple Factor
36 ÷ 2 = 18
Multiple Factor
18 ÷ 2 = 9
Multiple Factor
9 ÷ 3 = 3
Multiple Factor
3 ÷ 3 = 1
Multiple Factor
√72 = √2 × 2 × 2 × 3 × 3 = 2 × 3√2 = 6√2
2 is extracted from a pair of 2’s under the root sign.
3 is extracted from a pair of 3’s under the root sign.
an unpaired 2 is “left behind” under the root sign.
Question
What is √2 in decimal terms?
Answer
1.414213562
Answer Process
See Toggle.
√2 ⇔ 1.414213562
| Input | Display | Comment |
|
blinker | clears screen |
| 2 |
√2 | |
|
√2 | |
 |
1.414213562 | Answer (Toggle) |
Square Root |
Question
What is 6√2 in decimal terms?
Answer
8.485281374
Answer Process
See Toggle.
6√2 ⇔ 8.485281374
| Input | Display | Comment |
|
blinker | clears screen |
| 6 × 2 |
6*√2 | |
|
6√2 | |
|
8.485281374 | Answer (Toggle) |
Square Root |
Question
With reference to the question above, what is 6√2 in decimal terms rounded to the nearest hundredth?
Answer
8.49
Answer Process
See Rounding.
8.485281374 → 8.49
Practice – Questions
1. What is √49 ?
2. What is √2766.76 ?
3. Using the calculator, what is the simplified version of √108 ?
4. Without using the calculator, what is the simplified version of √108 ?
Show your work.
5. Using the calculator, what is the simplified version
of √5 × √15 ?
6. Without using the calculator, what is the simplified version of √5 × √15 ?
Show your work.
7. What is the simplified version of √48 ?
8. What is the simplified version of √6 × √14 ?
9. What is 4√3 in decimal terms?
10. With reference to Question 9, what is 4√3 in decimal terms rounded to the nearest tenth?
Practice – Answers
1. 7
2. 52.6
3. 6√3
4.
Factoring (Variation):
108 ÷ 2 = 54
Multiple Factor
54 ÷ 2 = 27
Multiple Factor
27 ÷ 3 = 9
Multiple Factor
9 ÷ 3 = 3
Multiple Factor
3 ÷ 3 = 1
Multiple Factor
√108 = √2 × 2 × 3 × 3 × 3 = 2 × 3√3 = 6√3
2 is extracted from a pair of 2’s under the root sign;
3 is extracted from a pair of 3’s under the root sign;
an unpaired 3 is “left behind” under the root sign.
5. 5√3
6.
√NumberA × √NumberB = √NumberA × NumberB
√5 × √15 = √5×15 = √75 = 5√3
Factoring (Variation):
75 ÷ 5 = 15
Multiple Factor
15 ÷ 5 = 3
Multiple Factor
3 ÷ 3 = 1
Multiple Factor
√75 = √5 × 5 × 3 = 5√3
5 is extracted from a pair of 5’s under the root sign;
an unpaired 3 is “left behind” under the root sign.
7. 4√3
8. 2√21
9. 6.92820323
10. 6.9