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Answer :
To simplify the expression
[tex]$$
(5xy^5)^2 (y^3)^4,
$$[/tex]
we follow these steps:
1. First, simplify the factor [tex]$\left(5xy^5\right)^2$[/tex]. When raising a product to a power, each term is raised to that power:
[tex]$$
(5xy^5)^2 = 5^2 \cdot x^2 \cdot (y^5)^2.
$$[/tex]
Calculate each part:
- [tex]$5^2 = 25$[/tex],
- [tex]$x^2$[/tex] remains as is,
- [tex]$(y^5)^2 = y^{5 \times 2} = y^{10}$[/tex].
So, we have:
[tex]$$
(5xy^5)^2 = 25x^2y^{10}.
$$[/tex]
2. Next, simplify the second factor [tex]$\left(y^3\right)^4$[/tex]. Raise [tex]$y^3$[/tex] to the power of 4:
[tex]$$
(y^3)^4 = y^{3 \times 4} = y^{12}.
$$[/tex]
3. Now multiply the results from the two steps:
[tex]$$
25x^2y^{10} \cdot y^{12}.
$$[/tex]
Multiply the powers of [tex]$y$[/tex] using the rule [tex]$y^a \cdot y^b = y^{a+b}$[/tex]:
[tex]$$
y^{10} \cdot y^{12} = y^{10+12} = y^{22}.
$$[/tex]
Therefore, the final expression is:
[tex]$$
25x^2y^{22}.
$$[/tex]
Thus, the correct simplification of the expression is
[tex]$$
\boxed{25x^2y^{22}}.
$$[/tex]
[tex]$$
(5xy^5)^2 (y^3)^4,
$$[/tex]
we follow these steps:
1. First, simplify the factor [tex]$\left(5xy^5\right)^2$[/tex]. When raising a product to a power, each term is raised to that power:
[tex]$$
(5xy^5)^2 = 5^2 \cdot x^2 \cdot (y^5)^2.
$$[/tex]
Calculate each part:
- [tex]$5^2 = 25$[/tex],
- [tex]$x^2$[/tex] remains as is,
- [tex]$(y^5)^2 = y^{5 \times 2} = y^{10}$[/tex].
So, we have:
[tex]$$
(5xy^5)^2 = 25x^2y^{10}.
$$[/tex]
2. Next, simplify the second factor [tex]$\left(y^3\right)^4$[/tex]. Raise [tex]$y^3$[/tex] to the power of 4:
[tex]$$
(y^3)^4 = y^{3 \times 4} = y^{12}.
$$[/tex]
3. Now multiply the results from the two steps:
[tex]$$
25x^2y^{10} \cdot y^{12}.
$$[/tex]
Multiply the powers of [tex]$y$[/tex] using the rule [tex]$y^a \cdot y^b = y^{a+b}$[/tex]:
[tex]$$
y^{10} \cdot y^{12} = y^{10+12} = y^{22}.
$$[/tex]
Therefore, the final expression is:
[tex]$$
25x^2y^{22}.
$$[/tex]
Thus, the correct simplification of the expression is
[tex]$$
\boxed{25x^2y^{22}}.
$$[/tex]
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