Sure, let's solve the problem step by step by expanding the given binomial expression [tex]\((x^{3n} + y^{3m})^3\)[/tex].
To expand [tex]\((a + b)^3\)[/tex], we use the binomial theorem, which is written as follows:
[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
In our case, we let:
[tex]\[
a = x^{3n} \quad \text{and} \quad b = y^{3m}
\][/tex]
Substituting these into our binomial expansion formula, we get:
[tex]\[
(x^{3n} + y^{3m})^3 = (x^{3n})^3 + 3(x^{3n})^2(y^{3m}) + 3(x^{3n})(y^{3m})^2 + (y^{3m})^3
\][/tex]
Now we need to simplify each term individually:
1. For [tex]\((x^{3n})^3\)[/tex]:
[tex]\[
(x^{3n})^3 = x^{3n \cdot 3} = x^{9n}
\][/tex]
2. For [tex]\(3(x^{3n})^2(y^{3m})\)[/tex]:
[tex]\[
3(x^{3n})^2(y^{3m}) = 3(x^{3n \cdot 2})(y^{3m}) = 3(x^{6n})(y^{3m}) = 3x^{6n}y^{3m}
\][/tex]
3. For [tex]\(3(x^{3n})(y^{3m})^2\)[/tex]:
[tex]\[
3(x^{3n})(y^{3m})^2 = 3(x^{3n})(y^{3m \cdot 2}) = 3(x^{3n})(y^{6m}) = 3x^{3n}y^{6m}
\][/tex]
4. For [tex]\((y^{3m})^3\)[/tex]:
[tex]\[
(y^{3m})^3 = y^{3m \cdot 3} = y^{9m}
\][/tex]
Combining all these terms together, we get:
[tex]\[
(x^{3n} + y^{3m})^3 = x^{9n} + 3x^{6n}y^{3m} + 3x^{3n}y^{6m} + y^{9m}
\][/tex]
So, the expanded form of [tex]\(\left(x^{3n} + y^{3m}\right)^3\)[/tex] is:
[tex]\[
x^{9n} + 3x^{6n}y^{3m} + 3x^{3n}y^{6m} + y^{9m}
\][/tex]
