root30+root30+root30+roit30=?
Answers
Answer:30
=
2
⋅
3
⋅
5
has no square factors, so it is not possible to simplify
√
30
.
You can calculate an approximation by hand as shown below...
Step-by-step explanation:I explained my favourite method (a sort of Newton Raphson method) for approximating square roots of integers in an answer to the following question:
How do you find the square root 28?
Given an integer
n
, choose integers
p
0
and
q
0
so
p
0
q
0
is a reasonable first approximation to
√
n
.
Then iterate using the formulas:
p
i
+
1
=
p
2
i
+
n
q
2
i
q
i
+
1
=
2
p
i
q
i
If the resulting values of
p
i
+
1
and
q
i
+
1
have a common factor, then divide both by that factor before the next iteration.
The successive pairs
p
i
,
q
i
provide a sequence of rational approximations
p
i
q
i
to
√
n
that converge quite rapidly.
For our example, let
n
=
30
,
p
0
=
11
,
q
0
=
2
(using an initial approximation of
5.5
since
30
is about halfway between
5
2
=
25
and
6
2
=
36
).
p
1
=
p
2
0
+
n
q
2
0
=
11
2
+
30
⋅
2
2
=
121
+
120
=
241
q
1
=
2
p
0
q
0
=
2
⋅
11
⋅
2
=
44
This would give
√
30
≈
241
44
=
5.47
.
7
.
2
p
2
=
p
2
1
+
n
q
2
1
=
241
2
+
30
⋅
44
2
=
58081
+
58080
=
116161
q
2
=
2
p
1
q
1
=
2
⋅
241
⋅
44
=
21208
This gives
√
30
≈
116161
21208
≈
5.477225575