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Square Roots
“ ^ “ plus any number equals exponent
“ { “ equals square root symbol
Perfect Squares
To find a perfect square of a number you take a whole
number that is between 1 and 10 and multiply it by itself.
Square of n = n^2
Square of 5=
5x5=
5^2=
25
The perfect squares are the squares of the whole numbers.
Perfect Squares:
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
Finding Square Roots of Numbers That Aren't
Perfect Squares Without a Calculator
To find the square root of a number without using a
calculator you must do the following steps:
1. Estimate - first, get as close as you can by finding two
perfect square roots your number is between.
2. Divide - divide your number by one of those square
roots.
3. Average - take the average of the result of step 2 and
the root.
4. Use the result of step 3 to repeat steps 2 and 3 until
you have a number that is accurate enough for you.
Example
Calculate the square root of 10 ({10) to 2 decimal places.
1. Find the two perfect square numbers it lies between.
Solution:
32 = 9 and 42 = 16, so {10 lies between 3 and 4.
2. Divide 10 by 3. 10/3 = 3.33
3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667
Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 +
3.1667)/2 = 3.1623
Check Your Answer - Is 3.1623 squared equal to 10?
3.1623 x 3.1623 = 10.0001
Practice Problems for Solving Square Roots
Without Using a Calculator
Problem#1:
60
Problem#2:
7
Problem#3:
75
Problem#4:
33
Problem#5:
66
Answer to Problem#1: 7.745
Answer to Problem#2: 2.645
Answer to Problem#3: 8.660
Answer to Problem#4: 5.744
Answer to Problem#5: 8.124
Properties of square roots
Square roots have the following properties
{a+b = {a + {b
{ab = {a * {b
{a/b = {a/{b
[{a]^2 = a
Addition and Subtraction of
Square Roots
Adding and subtracting square roots is just like
combining like terms when you need to do that with
algebraic expressions. If the indices (a square
root's index is 2, a cube root's index is 3, a 4th
root's index is 4, etc.) or the radicands (the
expression under the root sign or enclosed by
brackets after the square root) are the same.
When adding and subtracting you keep the bases
the same and add or subtract just the co-efficient.
Examples
4{2 + 7{2 = 11{2
12{3 – 8{3 = 4{3
2{2 – 5{2 + 67{2 = 64{2
Practice Problems Adding and Subtracting Square
Roots
Problem#1: 95{95 + 95{95=
Problem#2: 5{6 + 36{6=
Problem#3: 65{7 – 23{7=
Problem#4: 33{66 + 1{66 – 235{66=
Problem#5: 854{32 – 855{32 + {32=
Quick Solutions
Answer to Problem#1: 190{95
Answer to Problem#2: 41{6
Answer to Problem#3: 42{7
Answer to Problem#4: -201{66
Answer to Problem#5: 0
Multiplication and Division of Square Roots
To multiply a square root equation you multiply all of the
co-efficients together and then all of the radicals
together.
To divide a square root equation you have to take the two
terms and divide the co-efficients together and then all of
the radicals together.
Examples
2. 25{9/5{3=
25/5=5=
{9/{3={3=
5{3
2. 2{6*62{2=
2*62=124=
{6*{2={12
124{12
Practice Problems for Multiplication and
Division
Problem#1: 6{4*7{8=
Problem#2: 12{4/3{2=
Problem#3: 81{24/9{3=
Problem#4: 126{45*2{3=
Problem#5: 7{4^2=
Quick Solutions
Answer to Problem#1: 42{32
Answer to Problem#2: 36{8
Answer to Problem#3: 9{8
Answer to Problem#4: 63{15
Answer to Problem#5: 49{16
Mini Quiz
1. Is 75 a perfect square?
2. What is the root of 64?
3. {30=
4. {52=
5. 3{6 + 4{6=
6. 16{2 – 11{2=
7. 22{7*8{3=
8. 33{16/3{4=
9. 15{7^2=
10. 8{3^2 + 6{3^2=
Quick Solutions to Mini Quiz
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
No
8
5.477
7.211
7{6
5{2
176{21
11{4
30{7
100{9
Websites For Further Information
http://library.thinkquest.org/20991/textonly/alg/ro
ots.html?tqskip1=1&tqtime=0410
http://math.usask.ca/readin/roots.html
http://www.gomath.com/htdocs/lesson/radical_lesso
n3.htm
www.purple.com