how to solve the square root of 13 in step by step
Answers
Step-by-step explanation:
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Step-by-step explanation:
Use a Newton Raphson method to find:
√
13
≈
842401
233640
≈
3.60555127547
Explanation:
Since
13
is a prime number, there is no simpler form for its square root.
√
13
is an irrational number somewhere between
3
=
√
9
and
4
=
√
16
.
Linearly interpolating, a reasonable first approximation would be:
√
13
≈
3.6
=
18
5
We can get better approximations from our initial one (call it
a
0
) using a Newton Raphson method.
A typical formula used to derive a more accurate approximation for
√
n
would be:
a
i
+
1
=
a
2
i
+
n
2
a
i
I prefer to separate
a
i
into numerator
p
i
and denominator
q
i
. So
a
i
=
p
i
q
i
and we can iterate using the formulae:
{
p
i
+
1
=
p
2
i
+
n
q
2
i
q
i
+
1
=
2
p
i
q
i
In our example,
n
=
13
,
p
0
=
18
,
q
0
=
5
and we find:
{
p
1
=
p
2
0
+
13
q
2
0
=
324
+
13
⋅
25
=
649
q
1
=
2
p
0
q
0
=
180
If we stopped here our approximation would be:
√
13
≈
649
180
=
3.60
¯
5
Let's try one more iteration:
⎧
⎪
⎨
⎪
⎩
p
2
=
p
2
1
+
13
q
2
1
=
421201
+
13
⋅
32400
=
842401
q
2
=
2
p
1
q
1
=
233640
Stopping here, we have:
√
13
≈
842401
233640
≈
3.60555127547
Using a calculator:
√
13
≈
3.6055512754639