College

Choose the correct simplification of the expression [tex]\left(5 x y^5\right)^2\left(y^3\right)^4[/tex].

A. [tex]25 x^2 y^{22}[/tex]
B. [tex]10 x^2 y^{22}[/tex]
C. [tex]25 x^3 y^{14}[/tex]
D. [tex]10 x^3 y^{14}[/tex]

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].