Let's begin by expanding the given expression [tex]\((4x^3 + 7y^3z^4)^2\)[/tex].
The basic principle to apply here is the expansion of a binomial raised to a power, specifically using the square:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
For our problem, let’s set:
[tex]\[ a = 4x^3 \][/tex]
[tex]\[ b = 7y^3z^4 \][/tex]
Now we can apply the binomial expansion formula step-by-step:
1. Square the first term:
[tex]\[
(4x^3)^2 = 16(x^3)^2 = 16x^6
\][/tex]
2. Square the second term:
[tex]\[
(7y^3z^4)^2 = 49(y^3z^4)^2 = 49y^6z^8
\][/tex]
3. Multiply the two terms and then multiply by 2:
[tex]\[
2 \cdot (4x^3) \cdot (7y^3z^4) = 2 \cdot 28x^3y^3z^4 = 56x^3y^3z^4
\][/tex]
Now, we combine all these parts together:
[tex]\[
(4x^3 + 7y^3z^4)^2 = 16x^6 + 56x^3y^3z^4 + 49y^6z^8
\][/tex]
Thus, the expanded form of the expression [tex]\((4x^3 + 7y^3z^4)^2\)[/tex] is:
[tex]\[
16x^6 + 56x^3y^3z^4 + 49y^6z^8
\][/tex]
