express this in the form of a + ib {( 1/3) + 3i }
Answers
Answer:
1/The required form is (\frac{1}{3}+3i)^3=-\frac{242}{27}-26i(
3
1
+3i)
3
=−
27
242
−26i .
Step-by-step explanation:
Given : Expression (\frac{1}{3}+3i)^3(
3
1
+3i)
3
.
To find : Express the given complex number in the form a + ib ?
Solution :
Using algebraic identity to solve the expression,
(a+b)^3=a^3+b^3+3a^2b+3ab^2(a+b)
3
=a
3
+b
3
+3a
2
b+3ab
2
Here, a=\frac{1}{3},\ b=3ia=
3
1
, b=3i
Substitute the value,
(\frac{1}{3}+3i)^3=(\frac{1}{3})^3+(3i)^3+3(\frac{1}{3})^2(3i)+3(\frac{1}{3})(3i)^2(
3
1
+3i)
3
=(
3
1
)
3
+(3i)
3
+3(
3
1
)
2
(3i)+3(
3
1
)(3i)
2
(\frac{1}{3}+3i)^3=\frac{1}{27}+27i^3+i+9i^2(
3
1
+3i)
3
=
27
1
+27i
3
+i+9i
2
We know that, i^2=-1i
2
=−1
(\frac{1}{3}+3i)^3=\frac{1}{27}+27(-1)i+i+9(-1)(
3
1
+3i)
3
=
27
1
+27(−1)i+i+9(−1)
(\frac{1}{3}+3i)^3=\frac{1}{27}-27i+i-9(
3
1
+3i)
3
=
27
1
−27i+i−9
(\frac{1}{3}+3i)^3=\frac{1-243}{27}-26i(
3
1
+3i)
3
=
27
1−243
−26i
(\frac{1}{3}+3i)^3=-\frac{242}{27}-26i(
3
1
+3i)
3
=−
27
242
−26i
In form of a+ib the required answer is (\frac{1}{3}+3i)^3=-\frac{242}{27}-26i(
3
1
+3i)
3
=−
27
242
−26i
where, a=-\frac{242}{27}a=−
27
242
and b=-26