Answer :
To simplify the expression [tex]\((5xy^5)^2(y^3)^4\)[/tex], we'll perform the following steps:
1. Apply the Power Rule to the First Part [tex]\((5xy^5)^2\)[/tex]:
- The power rule states that when an exponent is applied to a product, you apply the exponent to each factor inside the parentheses.
- [tex]\((5xy^5)^2\)[/tex] can be expanded as:
- [tex]\(5^2 \times (x)^2 \times (y^5)^2\)[/tex]
- This becomes:
- [tex]\(25 \times x^2 \times y^{10}\)[/tex] since [tex]\(5^2 = 25\)[/tex] and [tex]\((y^5)^2 = y^{10}\)[/tex].
2. Apply the Power Rule to the Second Part [tex]\((y^3)^4\)[/tex]:
- Similarly, apply the exponent to the single term inside the parentheses.
- [tex]\((y^3)^4\)[/tex] becomes:
- [tex]\(y^{12}\)[/tex] because [tex]\((y^3)^4 = y^{3 \times 4} = y^{12}\)[/tex].
3. Combine the Results:
- Now that we have [tex]\(25 \times x^2 \times y^{10}\)[/tex] and [tex]\(y^{12}\)[/tex], we multiply these expressions together:
- The expression becomes:
- [tex]\(25 \times x^2 \times y^{10} \times y^{12}\)[/tex]
- Combine the [tex]\(y\)[/tex] terms using the rule [tex]\(y^a \times y^b = y^{a+b}\)[/tex]:
- [tex]\(y^{10} \times y^{12} = y^{22}\)[/tex]
4. Final Simplified Expression:
- The full simplified expression is [tex]\(25x^2y^{22}\)[/tex].
Therefore, the correct simplification is [tex]\(\boxed{25x^2y^{22}}\)[/tex].
1. Apply the Power Rule to the First Part [tex]\((5xy^5)^2\)[/tex]:
- The power rule states that when an exponent is applied to a product, you apply the exponent to each factor inside the parentheses.
- [tex]\((5xy^5)^2\)[/tex] can be expanded as:
- [tex]\(5^2 \times (x)^2 \times (y^5)^2\)[/tex]
- This becomes:
- [tex]\(25 \times x^2 \times y^{10}\)[/tex] since [tex]\(5^2 = 25\)[/tex] and [tex]\((y^5)^2 = y^{10}\)[/tex].
2. Apply the Power Rule to the Second Part [tex]\((y^3)^4\)[/tex]:
- Similarly, apply the exponent to the single term inside the parentheses.
- [tex]\((y^3)^4\)[/tex] becomes:
- [tex]\(y^{12}\)[/tex] because [tex]\((y^3)^4 = y^{3 \times 4} = y^{12}\)[/tex].
3. Combine the Results:
- Now that we have [tex]\(25 \times x^2 \times y^{10}\)[/tex] and [tex]\(y^{12}\)[/tex], we multiply these expressions together:
- The expression becomes:
- [tex]\(25 \times x^2 \times y^{10} \times y^{12}\)[/tex]
- Combine the [tex]\(y\)[/tex] terms using the rule [tex]\(y^a \times y^b = y^{a+b}\)[/tex]:
- [tex]\(y^{10} \times y^{12} = y^{22}\)[/tex]
4. Final Simplified Expression:
- The full simplified expression is [tex]\(25x^2y^{22}\)[/tex].
Therefore, the correct simplification is [tex]\(\boxed{25x^2y^{22}}\)[/tex].
