Answer:
√
8
≈
3
Explanation:
Note that:
2
2
=
4
<
8
<
9
=
3
2
Hence the (positive) square root of
8
is somewhere between
2
and
3
. Since
8
is much closer to
9
=
3
2
than
4
=
2
2
, we can deduce that the closest integer to the square root is
3
.
We can use this proximity of the square root of
8
to
3
to derive an efficient method for finding approximations.
Consider a quadratic with zeros
3
+
√
8
and
3
−
√
8
:
(
x
−
3
−
√
8
)
(
x
−
3
+
√
8
)
=
(
x
−
3
)
2
−
8
=
x
2
−
6
x
+
1
From this quadratic, we can define a sequence of integers recursively as follows:
⎧
⎪
⎨
⎪
⎩
a
0
=
0
a
1
=
1
a
n
+
2
=
6
a
n
+
1
−
a
n
The first few terms are:
0
,
1
,
6
,
35
,
204
,
1189
,
6930
,
...
The ratio between successive terms will tend very quickly towards
3
+
√
8
.
So:
√
8
≈
6930
1189
−
3
=
3363
1189
≈
2.828427
Coefficient of variation is the ratio of the standard deviation to the mean (average)
Coefficient of variation = (standard deviation/mean) × 100
×
=
Because 1/2 ≠ 1/6.
We know that 1/6 < 1/2, so we can set up an equation to see how many copies are needed for them to be equal.
(1/6)x = 1/2
[(1/6)x] × 6 = [1/2] × 6
x = 6/2 = 3
This equation shows that 1/6 × 3 = 1/2, therefore we need 3 copies of 1/6 to equal 1 copy of 1/2.
Answer:
Always
Step-by-step explanation:
Anything to the power 0 is 1
