Sure! Let's expand the expression [tex]\((2x^4 + 5y)^3\)[/tex] step-by-step.
To expand [tex]\((a + b)^3\)[/tex], we can use the binomial theorem, which states:
[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
In this case, [tex]\(a = 2x^4\)[/tex] and [tex]\(b = 5y\)[/tex].
Now we will substitute [tex]\(a = 2x^4\)[/tex] and [tex]\(b = 5y\)[/tex] into the binomial expansion formula.
1. Calculate [tex]\(a^3\)[/tex]:
[tex]\[
(2x^4)^3 = 2^3 \cdot (x^4)^3 = 8x^{12}
\][/tex]
2. Calculate [tex]\(3a^2b\)[/tex]:
[tex]\[
3 \cdot (2x^4)^2 \cdot 5y = 3 \cdot (2^2 \cdot (x^4)^2) \cdot 5y = 3 \cdot 4x^8 \cdot 5y = 60x^8y
\][/tex]
3. Calculate [tex]\(3ab^2\)[/tex]:
[tex]\[
3 \cdot (2x^4) \cdot (5y)^2 = 3 \cdot 2x^4 \cdot 25y^2 = 3 \cdot 50x^4y^2 = 150x^4y^2
\][/tex]
4. Calculate [tex]\(b^3\)[/tex]:
[tex]\[
(5y)^3 = 5^3 \cdot y^3 = 125y^3
\][/tex]
Now, combine all these terms:
[tex]\[
8x^{12} + 60x^8y + 150x^4y^2 + 125y^3
\][/tex]
So, the expanded form of the expression [tex]\(\left(2x^4 + 5y\right)^3\)[/tex] is:
[tex]\[
8x^{12} + 60x^8y + 150x^4y^2 + 125y^3
\][/tex]
