Sure, let's work through the details of squaring the expression [tex]\((4ab^2 + 5xy^3)\)[/tex].
1. Write down the expression to be squared:
[tex]\[
(4ab^2 + 5xy^3)^2
\][/tex]
2. Use the binomial theorem for squaring:
[tex]\[
(A + B)^2 = A^2 + 2AB + B^2
\][/tex]
Here, we can let [tex]\(A = 4ab^2\)[/tex] and [tex]\(B = 5xy^3\)[/tex].
3. Calculate each term separately:
[tex]\[
A^2 = (4ab^2)^2
\][/tex]
Squaring [tex]\(4ab^2\)[/tex]:
[tex]\[
(4ab^2)^2 = 16a^2b^4
\][/tex]
Next term:
[tex]\[
B^2 = (5xy^3)^2
\][/tex]
Squaring [tex]\(5xy^3\)[/tex]:
[tex]\[
(5xy^3)^2 = 25x^2y^6
\][/tex]
Lastly, the middle term:
[tex]\[
2AB = 2 \cdot (4ab^2) \cdot (5xy^3)
\][/tex]
Calculating the product:
[tex]\[
2 \cdot 4ab^2 \cdot 5xy^3 = 40ab^2xy^3 = 40ab^2x y^3
\][/tex]
4. Combine all terms:
[tex]\[
(4ab^2 + 5xy^3)^2 = A^2 + 2AB + B^2
\][/tex]
Substituting in the calculated values:
[tex]\[
16a^2b^4 + 40ab^2xy^3 + 25x^2y^6
\][/tex]
So, the expanded form of [tex]\((4ab^2 + 5xy^3)^2\)[/tex] is:
[tex]\[
16a^2b^4 + 40ab^2xy^3 + 25x^2y^6
\][/tex]
