How do you simplify −3−√23√17?
Answers
23 is a prime number, so it is not possible to simplify its square root, which is an irrational number a little less than
5
=
√
25
As such it is not expressible in the form
p
q
for integers
p
,
q
.
We can find rational approximations as follows:
23
=
5
2
−
2
is in the form
n
2
−
2
The square root of a number of the form
n
2
−
2
can be expressed as a continued fraction of standard form:
√
n
2
−
2
=
[
(
n
−
1
)
;
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
1
,
(
n
−
2
)
,
1
,
(
2
n
−
2
)
]
In our example
n
=
5
and we find:
√
23
=
[
4
;
¯¯¯¯¯¯¯¯¯¯¯¯¯¯
1
,
3
,
1
,
8
]
=
4
+
1
1
+
1
3
+
1
1
+
1
8
+
1
1
+
1
3
+
1
1
+
...
To use this to derive a good approximation for
√
23
terminate it early, just before one of the
8
's. For example:
√
23
≈
[
4
;
1
,
3
,
1
,
8
,
1
,
3
,
1
]
=
4
+
1
1
+
1
3
+
1
1
+
1
8
+
1
1
+
1
3
+
1
1
=
1151
240
=
4.7958
¯
3
With a calculator, we find:
√
23
≈
4.79583152
So our approximation is not bad.