If y=
3
−
2
, then find the value of (
y
3
1
−y
3
).
- A11
- B11\sqrt{2}
- C22
- D22\sqrt{2}
Solution & Step-by-step Explanation
Given:
y=
3
−
2
First, let's find
y
1
by rationalizing the denominator:
y
1
=
3
−
2
1
=
(
3
−
2
)(
3
+
2
)
3
+
2
=
3−2
3
+
2
=
3
+
2
Now, find the value of
y
1
−y:
y
1
−y=(
3
+
2
)−(
3
−
2
)=2
2
We need to find
y
3
1
−y
3
. Using the identity x
3
−y
3
=(x−y)
3
+3xy(x−y), let x=
y
1
and y=y:
y
3
1
−y
3
=(
y
1
−y)
3
+3(
y
1
)(y)(
y
1
−y)
y
3
1
−y
3
=(2
2
)
3
+3(2
2
)
(2
2
)
3
=8×2
2
=16
2
y
3
1
−y
3
=16
2
+6
2
=22
2
y=
3
−
2
First, let's find
y
1
by rationalizing the denominator:
y
1
=
3
−
2
1
=
(
3
−
2
)(
3
+
2
)
3
+
2
=
3−2
3
+
2
=
3
+
2
Now, find the value of
y
1
−y:
y
1
−y=(
3
+
2
)−(
3
−
2
)=2
2
We need to find
y
3
1
−y
3
. Using the identity x
3
−y
3
=(x−y)
3
+3xy(x−y), let x=
y
1
and y=y:
y
3
1
−y
3
=(
y
1
−y)
3
+3(
y
1
)(y)(
y
1
−y)
y
3
1
−y
3
=(2
2
)
3
+3(2
2
)
(2
2
)
3
=8×2
2
=16
2
y
3
1
−y
3
=16
2
+6
2
=22
2