Square Root

Square Root =  Number² = √ Number × Itself 


Basics
Square Root = Number² = √Number × Itself

Square Root of 9 = 9 =  = √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.
53  =  5 × √3
a3  =  a × √3

2nd square root is the square root command.
2nd activates square root above square .

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 provided by Mathematics Formula Sheet)

Square Root = Number² = √Number × Itself

NumberA × √NumberB = √NumberA × NumberB


Question
What is 16 ?

Answer
4

Calculator Click What You See Comment
clear blinker clears screen
2nd square root 16 16 Square Root
enter 4 Answer
Square Root


Question

What is 5097.96 ?

Answer
71.4

Calculator Click What You See Comment
clear blinker clears screen
2nd square root 5097.96 5097.96 Square Root
enter 71.4 Answer
Square Root


Question

Using the calculator, what is the simplified version of 180 ?

Answer
65

Calculator Click What You See Comment
clear blinker clears screen
2nd square root 180 180 Square Root
enter 65 Answer
Square Root


Question

Without using the calculator, what is the simplified version of 180 ?

Answer
65

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
62

Calculator Click What You See Comment
clear blinker clears screen
2nd square rootright × 2nd square root 12 6  × √12
enter 6√2 Answer
Square Root


Question
Without using the calculator, what is the simplified version of √6 × √12 ?

Answer
62

Answer Process
NumberA × √NumberB = √NumberA × NumberB
6 × √12 = √6×12 = √72 = 62

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 × 32  =  62
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

Calculator Click What You See Comment
clear blinker clears screen
2nd square root 2 2
enter 2
toggle 1.414213562 Answer
(Toggle)
Square Root


Question

What is 62 in decimal terms?

Answer
8.485281374

Answer Process
See Toggle.
6√2  ⇔ 8.485281374

Calculator Click What You See Comment
clear blinker clears screen
6 × 2nd square root 2 6*√2
enter 6√2
toggle 8.485281374 Answer
(Toggle)
Square Root


Question
With reference to the question above, what is 62 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. Without using the calculator, what is the simplified version of 48 ?

8. Without using the calculator, what is the simplified version of √6 × √14  ?

9.  What is 43 in decimal terms?

10.  With reference to Question 9, what is 43 in decimal terms rounded to the nearest tenth?


Practice – Answers
1.  7

2.  52.6

3.  63

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.  53

6.
NumberA × √NumberB = √NumberA × NumberB
5 × √15 = √5×15 = √75 = 53

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.  43

8.  221

9.  6.92820323

10.  6.9

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