Answer :
Let's simplify the expression [tex]\((5xy^5)^2(y^3)^4\)[/tex] step by step.
1. Simplify [tex]\((5xy^5)^2\)[/tex]:
- The expression [tex]\((5xy^5)^2\)[/tex] involves raising each part inside the parentheses to the power of 2.
- For the number 5: [tex]\((5)^2 = 25\)[/tex].
- For [tex]\(x\)[/tex]: Since it's [tex]\(x^1\)[/tex], [tex]\((x^1)^2 = x^{1 \times 2} = x^2\)[/tex].
- For [tex]\(y^5\)[/tex]: [tex]\((y^5)^2 = y^{5 \times 2} = y^{10}\)[/tex].
So, [tex]\((5xy^5)^2 = 25x^2y^{10}\)[/tex].
2. Simplify [tex]\((y^3)^4\)[/tex]:
- For [tex]\(y^3\)[/tex], raise it to the power of 4: [tex]\((y^3)^4 = y^{3 \times 4} = y^{12}\)[/tex].
3. Combine the results:
Now we need to multiply the expressions we simplified:
[tex]\[
25x^2y^{10} \times y^{12}
\][/tex]
- Coefficients: The number is already 25.
- For [tex]\(x\)[/tex]: There is no x in the second expression, so it remains [tex]\(x^2\)[/tex].
- For [tex]\(y\)[/tex]: Combine the exponents as they have the same base:
[tex]\[
y^{10} \times y^{12} = y^{10 + 12} = y^{22}
\][/tex]
Putting it all together, the expression simplifies to:
[tex]\[
25x^2y^{22}
\][/tex]
Therefore, the correct simplification of the expression is [tex]\(25x^2y^{22}\)[/tex].
1. Simplify [tex]\((5xy^5)^2\)[/tex]:
- The expression [tex]\((5xy^5)^2\)[/tex] involves raising each part inside the parentheses to the power of 2.
- For the number 5: [tex]\((5)^2 = 25\)[/tex].
- For [tex]\(x\)[/tex]: Since it's [tex]\(x^1\)[/tex], [tex]\((x^1)^2 = x^{1 \times 2} = x^2\)[/tex].
- For [tex]\(y^5\)[/tex]: [tex]\((y^5)^2 = y^{5 \times 2} = y^{10}\)[/tex].
So, [tex]\((5xy^5)^2 = 25x^2y^{10}\)[/tex].
2. Simplify [tex]\((y^3)^4\)[/tex]:
- For [tex]\(y^3\)[/tex], raise it to the power of 4: [tex]\((y^3)^4 = y^{3 \times 4} = y^{12}\)[/tex].
3. Combine the results:
Now we need to multiply the expressions we simplified:
[tex]\[
25x^2y^{10} \times y^{12}
\][/tex]
- Coefficients: The number is already 25.
- For [tex]\(x\)[/tex]: There is no x in the second expression, so it remains [tex]\(x^2\)[/tex].
- For [tex]\(y\)[/tex]: Combine the exponents as they have the same base:
[tex]\[
y^{10} \times y^{12} = y^{10 + 12} = y^{22}
\][/tex]
Putting it all together, the expression simplifies to:
[tex]\[
25x^2y^{22}
\][/tex]
Therefore, the correct simplification of the expression is [tex]\(25x^2y^{22}\)[/tex].
