Find the value of 27a
3
+
a
3

1
​
, if 9a
2
+
a
2

1
​
=43.

We are given:
9a
2
+
a
2

1
​
=43
We can rewrite this as:

(3a)
2
+(
a
1
​
)
2
=43
Using the identity x
2
+y
2
=(x+y)
2
−2xy, let x=3a and y=
a
1
​
:

(3a+
a
1
​
)
2
−2(3a)(
a
1
​
)=43
(3a+
a
1
​
)
2
−6=43
(3a+
a
1
​
)
2
=49
3a+
a
1
​
=7(taking the positive root for standard evaluation)
Now, we need to find the value of 27a
3
+
a
3

1
​
, which is (3a)
3
+(
a
1
​
)
3
.
Using the identity x
3
+y
3
=(x+y)
3
−3xy(x+y):

(3a)
3
+(
a
1
​
)
3
=(3a+
a
1
​
)
3
−3(3a)(
a
1
​
)(3a+
a
1
​
)
27a
3
+
a
3

1
​
=(7)
3
−3(3)(7)
27a
3
+
a
3

1
​
=3443−63=280