Answer :
Answer:
3Touch and hold a clip to pin it. Unpinned clips will be deleted after 1 hour.Tap on a clip to paste it in the text box.Welcome to Gboard clipboard, any text you copy will be saved here.Use the edit icon to pin, add or delete clips.1Certainly! Let's prove that
3
+
3
=
2
x
3
+3bx=2a given
=
(
+
2
+
3
)
1
/
3
+
(
−
2
+
3
)
1
/
3
x=(a+
a
2
+b
3
)
1/3
+(a−
a
2
+b
3
)
1/3
.
### Step-by-Step Solution:
1. Define the Variables:
Let
=
(
+
2
+
3
)
1
/
3
m=(a+
a
2
+b
3
)
1/3
and
=
(
−
2
+
3
)
1
/
3
n=(a−
a
2
+b
3
)
1/3
.
2. Express
x in Terms of
m and
n :
Therefore, we have
=
+
x=m+n .
3. Cube
x :
We are asked to find
3
x
3
. Using the binomial theorem for
(
+
)
3
(m+n)
3
, we get:
3
=
(
+
)
3
=
3
+
3
+
3
(
+
)
x
3
=(m+n)
3
=m
3
+n
3
+3mn(m+n)
4. Compute
3
m
3
and
3
n
3
:
From the definitions,
3
=
+
2
+
3
m
3
=a+
a
2
+b
3
and
3
=
−
2
+
3
n
3
=a−
a
2
+b
3
.
5. Sum
3
m
3
and
3
n
3
:
3
+
3
=
(
+
2
+
3
)
+
(
−
2
+
3
)
=
2
m
3
+n
3
=(a+
a
2
+b
3
)+(a−
a
2
+b
3
)=2a
6. Determine
mn :
To compute
mn :
=
(
(
+
2
+
3
)
1
/
3
)
(
(
−
2
+
3
)
1
/
3
)
mn=((a+
a
2
+b
3
)
1/3
)((a−
a
2
+b
3
)
1/3
)
Notice:
=
(
(
+
2
+
3
)
(
−
2
+
3
)
)
1
/
3
=
(
2
−
(
2
+
3
)
2
)
1
/
3
=
(
2
−
(
2
+
3
)
)
1
/
3
=
(
−
3
)
1
/
3
=
−
mn=((a+
a
2
+b
3
)(a−
a
2
+b
3
))
1/3
=(a
2
−(
a
2
+b
3
)
2
)
1/3
=(a
2
−(a
2
+b
3
))
1/3
=(−b
3
)
1/3
=−b
7. Substitute Values in the Expansion:
Recall the expansion
3
=
3
+
3
+
3
(
+
)
x
3
=m
3
+n
3
+3mn(m+n) :
Using
+
=
m+n=x ,
3
+
3
=
2
m
3
+n
3
=2a , and
=
−
mn=−b , it follows that:
3
=
2
+
3
(
−
)
x
3
=2a+3(−b)x
8. Result:
Rearrange the terms:
3
+
3
=
2
x
3
+3bx=2a
Thus, we have proved that:
3
+
3
=
2
x
3
+3bx=2a
as required.
