Calculate
Square Root . .
.
without using a
square root
calculator
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Example
of
Direct
Calculation
Precise,
digit by digit
method used to
Calculate
Square Roots
(The Chinese
Method).
Video
demonstrating
this
method 
click
here
Calculate
Square
Root:
Step
1:
Beginning at
the decimal
point,
divide
the radicand
into
groups of
two digits
in both
directions. The
decimal can be
extended as far
as you
wish.
Step
2:
Beginning on
the left,
select
the first
group of
one or
two
digits.
In this
example
, the
first
group of
digits
in the radicand
is simply the
number
2
.
Step
3:
If
possible,
find
the perfect
square root
(expressed as
an integer) of
the first
group of
digits in
the
radicand.
Or,
If the
first group of
digits does not
have a perfect
square
root
(expressed as
an integer),
find the
perfect
square root of
the number
closest to, but
less than, the
first group of
digits.
In this
example
, the
number 2 does
not have a
perfect square
root
(which can be
expressed as an
integer).
The
number closest
to, but less
than, the
number 2 which
has a perfect
square
root is
the number
1
.
The
perfect square
root of the
number 1
is also
1
. Write this
number above
the radical
sign as shown
below.
Step
4:
Square
the answer to
step 3, and
subtract
it from the
first group of
numbers.
In this
example
, the
number 1
squared is
subtracted from
the number
2 .
Step
5:
 A

Bring
down
the
next
group
of
two
digits.
In this
example,
bring down
73.
 B

Double
the
result
of
step
3
(shown
as a
slate
color),
and
write
on
lower
left.
In this
example,
the number
1 x 2 =
2.
 C
 Add
another
digit
to
the
number
computed
in
"B"
(the
slate
colored
number)
on
the
lower
left.
Choose
the
new
digit
so
that
the
new
number
created
by
the
additional
digit
can
be
divided
into
the
number
on
the
immediate
right
the
same
number
of
times
as
the
new
digit.
In this
example,
the new
digit is
6. The
resulting
number
(26), can
be divided
into 173 a
total of 6
times.
Step
6:
Repeat
steps A, B, and
C shown in step
5.
 A

Bring
down
the
next
group
of
two
digits.
In this
example,
bring down
00.
 B

Double
the
result
of
step
5
(shown
as a
slate
color),
and
write
on
lower
left.
In this
example,
the number
16 x 2 =
32.
 C
 Add
another
digit
to
the
number
computed
in
"B"
(the
slate
colored
number)
on
the
lower
left.
Choose
the
new
digit
so
that
the
new
number
created
by
the
additional
digit
can
be
divided
into
the
number
on
the
immediate
right
the
same
number
of
times
as
the
new
digit.
In this
example,
the new
digit is
5. The
resulting
number
(325), can
be divided
into 1700
a total of
5
times.
Step
7:
Repeat
steps A, B, and
C shown in step
5.
 A

Bring
down
the
next
group
of
two
digits.
In this
example,
bring down
00.
 B

Double
the
result
of
step
6
(shown
as a
slate
color),
and
write
on
lower
left.
In this
example,
the number
165 x 2 =
330.
 C
 Add
another
digit
to
the
number
computed
in
"B"
(the
slate
colored
number)
on
the
lower
left.
Choose
the
new
digit
so
that
the
new
number
created
by
the
additional
digit
can
be
divided
into
the
number
on
the
immediate
right
the
same
number
of
times
as
the
new
digit.
In this
example,
the new
digit is
2. The
resulting
number
(3302),
can be
divided
into 7500
a total of
2
times.
Step
8:
Repeat
steps A, B, and
C shown in step
5.
 A

Bring
down
the
next
group
of
two
digits.
In this
example,
bring down
00.
 B

Double
the
result
of
step
7
(shown
as a
slate
color),
and
write
on
lower
left.
In this
example,
the number
1652 x 2 =
3304.
 C
 Add
another
digit
to
the
number
computed
in
"B"
(the
slate
colored
number)
on
the
lower
left.
Choose
the
new
digit
so
that
the
new
number
created
by
the
additional
digit
can
be
divided
into
the
number
on
the
immediate
right
the
same
number
of
times
as
the
new
digit.
In this
example,
the new
digit is
2. The
resulting
number
(33042),
can be
divided
into 89600
a total of
2
times.
The
final answer
is:
to 3
decimal
points
(not
rounded)
Cube
Roots:
calculating
cube roots
by hand is
more
complicated
than
computing
square roots
by hand. For
information
on cube
roots, see
the
following
web sites:
http://mathforum.org/library/drmath/view/52605.html
http://www.mathpath.org/Algor/cuberoot/algor.cube.root.htm
Example
of
Newton's
Method
NewtonRaphson
method:
consecutive
numerical
approximations
used to
Calculate
Square Roots
(approximations
become more and
more accurate
with each
iteration).
The three steps
used to
Calculate
Square Roots
are:
 A

Initial
Estimate
of
the
square
root
of
the
radicand.
(The
initial
estimate
can
be
any
number.
Subsequent
steps
will
refine
the
estimate.)
 B

Divide
the
radicand
by
the
estimate
in
part
A.
 C

refine
the
estimate
by
computing
the
Average
of
the
estimate
(from
part
A)
and
the
quotient
(from
part
B).
Video
demonstrating
this
method 
click
here
Calculate
Square
Root:
1st
Approximation:
Calculate
Square Root 
Newton's
Method
 A

Initial
Estimate
of
the
square
root
of
the
radicand
In this
example,
a
good
initial
estimate
for the
square
root of
273
is
15
.
(However,
the
initial
estimate
could be
"any"
number.
Subsequent
steps will
refine the
estimate.)
 B

Divide
the
radicand
by
the
estimate
in
part
A.
In this
example,
divide 273
by
15.
 C

1st
Approximation

refine
the
initial
estimate
by
computing
the
Average
of
the
estimate
(from
part
A)
and
the
quotient
(from
part
B).
In this
example,
, find
the
average of
15 and
18.2
.
1st
Approximation
of Square
Root is:
2nd
approximation:
 A

Initial
Estimate
for
the
2nd
Approximation
of
the
square
root
of
the
radicand.
Use
the
results
of
the
1st
Approximation,
part
C,
above.
In this
example,
the
initial
estimate
will
be
16.6
 B

Divide
the
radicand
by
the
estimate
in
part
A.
In this
example,
divide 273
by 16.6
.
 C

2nd
Approximation

refine
the
initial
estimate
by
computing
the
Average
of
the
estimate
(from
part
A)
and
the
quotient
(from
part
B).
In this
example,
, find
the
average of
16.6 and
16.445783
.
2nd
Approximation
of Square
Root is:
3rd
approximation:
 A

Initial
Estimate
for
the
3rd
Approximation
of
the
square
root
of
the
radicand.
Use
the
results
of
the
2nd
Approximation,
part
C,
above.
In this
example,
the
initial
estimate
will
be
16.522892
 B

Divide
the
radicand
by
the
estimate
in
part
A.
In this
example,
divide 273
by
16.522892
.
 C

3rd
Approximation

refine
the
initial
estimate
by
computing
the
Average
of
the
estimate
(from
part
A)
and
the
quotient
(from
part
B).
In this
example,
, find
the
average of
16.522892
and
16.522531
.
3rd
Approximation
of Square
Root is:
(Final
answer for
this
example)
Example
of the
Guess
& Check
Method
Using the
Guess & Check
Method to
Calculate
Square Roots is
exactly what
the name
implies: Guess
what the value
of the square
root might be,
then multiply
your guess by
itself to see
if the product
is equal to the
original
radicand.
The four steps
used to
Calculate
Square Roots
are:
 A

Guess

Estimate
the
square
root
of
the
radicand.
(The
initial
estimate
can
be
any
number.
Subsequent
steps
will
refine
the
estimate.)
 B

Multiply
the
estimate
by
itself.
 C

Check

Compare
the
product
obtained
in
step
"B"
(above)
with
the
radicand.
 D

Refine
the
Estimate
. If
the
product
obtained
in
step
"B"
(above)
is
greater
than
the
radicand,
decrease
the
estimate.
If
the
product
obtained
in
step
"B"
(above)
is
less
than
the
radicand,
increase
the
estimate.
 E

Repeat
steps
B
through
D
until
the
estimate
is an
accurate
square
root
of
the
radicand.
Calculate
Square
Root:
1st
Approximation:
Calculate
Square Root 
Guess & Check
Method
 A

Guess

Initial
Estimate
of
the
square
root
of
the
radicand.
In this
example,
, a good
initial
estimate
for the
square
root of
273 is
15
(However,
the
initial
estimate
could be
"any"
number.)
 B

Multiply
the
estimate
by
itself.
In this
example
,
multiply
15 by
15
.
 C

Check

Compare
the
product
obtained
in
step
"B"
(above)
with
the
radicand.
In this
example,
225 is
less than
273.
 D

Refine
the
Estimate
. If
the
product
obtained
in
step
"B"
(above)
is
greater
than
the
radicand,
decrease
the
estimate.
If
the
product
obtained
in
step
"B"
(above)
is
less
than
the
radicand,
increase
the
estimate.
In this
example,
since 225
is less
than 273
the
estimate
will be
increased
from 15 to
16
.
Refined
estimate of
Square
Root is:
2nd
Approximation:
Calculate
Square Root 
Guess & Check
Method
 A

Guess

Initial
Estimate
of
the
square
root
of
the
radicand.
Use
the
results
of
the
1st
Approximation,
part
D,
above.
In this
example
, the
estimate
for the
square
root of
273 has
been
raised to
16
 B

Multiply
the
estimate
by
itself.
In this
example
,
multiply
16 by
16.
 C

Check

Compare
the
product
obtained
in
step
"B"
(above)
with
the
radicand.
In this
example,
256 is
still less
than
273.
 D

Refine
the
Estimate
. If
the
product
obtained
in
step
"B"
(above)
is
greater
than
the
radicand,
decrease
the
estimate.
If
the
product
obtained
in
step
"B"
(above)
is
less
than
the
radicand,
increase
the
estimate.
In this
example
, since
256 is
less than
273 (but
closer
than the
original
estimate)
the
estimate
will be
increased
a small
amount,
from 16 to
16.5
.
Refined
estimate of
Square
Root is:
3rd
Approximation:
Calculate
Square Root 
Guess & Check
Method
 A

Guess

Initial
Estimate
of
the
square
root
of
the
radicand.
Use
the
results
of
the
2nd
Approximation,
part
D,
above.
In this
example
, the
estimate
for the
square
root of
273 has
been
raised to
16.5
 B

Multiply
the
estimate
by
itself.
In this
example
,
multiply
16.5 by
16.5
 C

Check

Compare
the
product
obtained
in
step
"B"
(above)
with
the
radicand.
In this
example,
272.25 is
still less
than
273
, but it
is much
closer
than the
other
estimates.
 D

Refine
the
Estimate
. If
the
product
obtained
in
step
"B"
(above)
is
greater
than
the
radicand,
decrease
the
estimate.
If
the
product
obtained
in
step
"B"
(above)
is
less
than
the
radicand,
increase
the
estimate.
In this
example
, since
272.25 is
less than
273 (but
much
closer
than the
other two
estimates)
the
estimate
should be
increased
slightly
.
Refined
estimate of
Square
Root so
far is: