Answer :
To solve this problem, we need to simplify both expressions provided, [tex]\(4 x^2 \sqrt{5 x^4}\)[/tex] and [tex]\(3 \sqrt{5 x^6}\)[/tex], and determine if they are equivalent.
### Simplifying [tex]\(4 x^2 \sqrt{5 x^4}\)[/tex]:
1. Inside the square root:
[tex]\[\sqrt{5 x^4}\][/tex]
This can be split into:
[tex]\[\sqrt{5} \cdot \sqrt{x^4}\][/tex]
2. Simplify [tex]\(\sqrt{x^4}\)[/tex]:
[tex]\[\sqrt{x^4} = x^2\][/tex]
3. Substitute back:
[tex]\[\sqrt{5 x^4} = \sqrt{5} \cdot x^2\][/tex]
4. Multiply with the coefficient [tex]\(4 x^2\)[/tex]:
[tex]\(4 x^2 \cdot (\sqrt{5} \cdot x^2)\)[/tex]:
[tex]\[4 x^2 \cdot x^2 \cdot \sqrt{5} = 4 x^4 \sqrt{5}\][/tex]
### Simplifying [tex]\(3 \sqrt{5 x^6}\)[/tex]:
1. Inside the square root:
[tex]\[\sqrt{5 x^6}\][/tex]
This can be split into:
[tex]\[\sqrt{5} \cdot \sqrt{x^6}\][/tex]
2. Simplify [tex]\(\sqrt{x^6}\)[/tex]:
[tex]\[\sqrt{x^6} = x^3\][/tex]
3. Substitute back:
[tex]\[\sqrt{5 x^6} = \sqrt{5} \cdot x^3\][/tex]
4. Multiply with the coefficient [tex]\(3\)[/tex]:
[tex]\(3 \cdot (\sqrt{5} \cdot x^3)\)[/tex]:
[tex]\[3 x^3 \sqrt{5}\][/tex]
### Compare the two simplified expressions:
- The first expression simplifies to [tex]\(4 x^4 \sqrt{5}\)[/tex]
- The second expression simplifies to [tex]\(3 x^3 \sqrt{5}\)[/tex]
It is clear that the two expressions are not equivalent. Therefore, none of the provided expressions are equivalent to both [tex]\(4 x^2 \sqrt{5 x^4}\)[/tex] and [tex]\(3 \sqrt{5 x^6}\)[/tex] simultaneously.
However, we are asked for the equivalent expression, where only one of the given options likely consider one of the above equivalence results only (if it is matching).
### Considering both provided simplified results with the Options:
- A. [tex]\(12 x^{10} \sqrt{5}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- B. [tex]\(60 x^8\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- C. [tex]\(35 x^{18}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- D. [tex]\(7 x^{10} \sqrt{5}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
Since no option matches directly any simplified term:
Thus, none of the options accurately represent the equivalence condition of the given mathematical problem based on simplifications.
### Simplifying [tex]\(4 x^2 \sqrt{5 x^4}\)[/tex]:
1. Inside the square root:
[tex]\[\sqrt{5 x^4}\][/tex]
This can be split into:
[tex]\[\sqrt{5} \cdot \sqrt{x^4}\][/tex]
2. Simplify [tex]\(\sqrt{x^4}\)[/tex]:
[tex]\[\sqrt{x^4} = x^2\][/tex]
3. Substitute back:
[tex]\[\sqrt{5 x^4} = \sqrt{5} \cdot x^2\][/tex]
4. Multiply with the coefficient [tex]\(4 x^2\)[/tex]:
[tex]\(4 x^2 \cdot (\sqrt{5} \cdot x^2)\)[/tex]:
[tex]\[4 x^2 \cdot x^2 \cdot \sqrt{5} = 4 x^4 \sqrt{5}\][/tex]
### Simplifying [tex]\(3 \sqrt{5 x^6}\)[/tex]:
1. Inside the square root:
[tex]\[\sqrt{5 x^6}\][/tex]
This can be split into:
[tex]\[\sqrt{5} \cdot \sqrt{x^6}\][/tex]
2. Simplify [tex]\(\sqrt{x^6}\)[/tex]:
[tex]\[\sqrt{x^6} = x^3\][/tex]
3. Substitute back:
[tex]\[\sqrt{5 x^6} = \sqrt{5} \cdot x^3\][/tex]
4. Multiply with the coefficient [tex]\(3\)[/tex]:
[tex]\(3 \cdot (\sqrt{5} \cdot x^3)\)[/tex]:
[tex]\[3 x^3 \sqrt{5}\][/tex]
### Compare the two simplified expressions:
- The first expression simplifies to [tex]\(4 x^4 \sqrt{5}\)[/tex]
- The second expression simplifies to [tex]\(3 x^3 \sqrt{5}\)[/tex]
It is clear that the two expressions are not equivalent. Therefore, none of the provided expressions are equivalent to both [tex]\(4 x^2 \sqrt{5 x^4}\)[/tex] and [tex]\(3 \sqrt{5 x^6}\)[/tex] simultaneously.
However, we are asked for the equivalent expression, where only one of the given options likely consider one of the above equivalence results only (if it is matching).
### Considering both provided simplified results with the Options:
- A. [tex]\(12 x^{10} \sqrt{5}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- B. [tex]\(60 x^8\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- C. [tex]\(35 x^{18}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
- D. [tex]\(7 x^{10} \sqrt{5}\)[/tex]:
[tex]\(\text{Clearly not matching as terms hence not equivalent, so ignore.}\)[/tex]
Since no option matches directly any simplified term:
Thus, none of the options accurately represent the equivalence condition of the given mathematical problem based on simplifications.
