Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To expand the expression [tex]\(\left(3 a^2 b^2 + 5 x^2 y^2\right)^2\)[/tex], we can use the binomial theorem, which states that [tex]\((u + v)^2 = u^2 + 2uv + v^2\)[/tex].
Here, [tex]\(u = 3 a^2 b^2\)[/tex] and [tex]\(v = 5 x^2 y^2\)[/tex].
Let's expand the expression step-by-step:
1. Calculate [tex]\(u^2\)[/tex]:
[tex]\[ u^2 = \left(3 a^2 b^2\right)^2 = \left(3\right)^2 \left(a^2\right)^2 \left(b^2\right)^2 = 9 a^4 b^4 \][/tex]
2. Calculate [tex]\(2uv\)[/tex]:
[tex]\[ 2uv = 2 \times (3 a^2 b^2) \times (5 x^2 y^2) = 2 \times 3 \times 5 \times a^2 b^2 \times x^2 y^2 = 30 a^2 b^2 x^2 y^2 \][/tex]
3. Calculate [tex]\(v^2\)[/tex]:
[tex]\[ v^2 = \left(5 x^2 y^2\right)^2 = \left(5\right)^2 \left(x^2\right)^2 \left(y^2\right)^2 = 25 x^4 y^4 \][/tex]
Now, combine these results to form the expanded expression:
[tex]\[ (u + v)^2 = u^2 + 2uv + v^2 \][/tex]
Substitute the expressions [tex]\(u^2\)[/tex], [tex]\(2uv\)[/tex], and [tex]\(v^2\)[/tex] back into the formula:
[tex]\[ (3 a^2 b^2 + 5 x^2 y^2)^2 = 9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4 \][/tex]
So, the expanded form of [tex]\(\left(3 a^2 b^2 + 5 x^2 y^2\right)^2\)[/tex] is:
[tex]\[ \boxed{9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4} \][/tex]
Here, [tex]\(u = 3 a^2 b^2\)[/tex] and [tex]\(v = 5 x^2 y^2\)[/tex].
Let's expand the expression step-by-step:
1. Calculate [tex]\(u^2\)[/tex]:
[tex]\[ u^2 = \left(3 a^2 b^2\right)^2 = \left(3\right)^2 \left(a^2\right)^2 \left(b^2\right)^2 = 9 a^4 b^4 \][/tex]
2. Calculate [tex]\(2uv\)[/tex]:
[tex]\[ 2uv = 2 \times (3 a^2 b^2) \times (5 x^2 y^2) = 2 \times 3 \times 5 \times a^2 b^2 \times x^2 y^2 = 30 a^2 b^2 x^2 y^2 \][/tex]
3. Calculate [tex]\(v^2\)[/tex]:
[tex]\[ v^2 = \left(5 x^2 y^2\right)^2 = \left(5\right)^2 \left(x^2\right)^2 \left(y^2\right)^2 = 25 x^4 y^4 \][/tex]
Now, combine these results to form the expanded expression:
[tex]\[ (u + v)^2 = u^2 + 2uv + v^2 \][/tex]
Substitute the expressions [tex]\(u^2\)[/tex], [tex]\(2uv\)[/tex], and [tex]\(v^2\)[/tex] back into the formula:
[tex]\[ (3 a^2 b^2 + 5 x^2 y^2)^2 = 9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4 \][/tex]
So, the expanded form of [tex]\(\left(3 a^2 b^2 + 5 x^2 y^2\right)^2\)[/tex] is:
[tex]\[ \boxed{9 a^4 b^4 + 30 a^2 b^2 x^2 y^2 + 25 x^4 y^4} \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.