Answer :
To determine which of the given expressions is a like radical to [tex]\( 3 \times \sqrt{5} \)[/tex], we need to understand what a "like radical" is. Like radicals are expressions that have the same radicand (the number under the radical sign) and the same index (the type of root being taken, such as square root or cube root).
Let's analyze each option step by step:
### Option 1: [tex]\( x \times \sqrt[3]{5} \)[/tex]
- This option involves the cube root of [tex]\( 5 \)[/tex]. The expression [tex]\(\sqrt[3]{5}\)[/tex] is a cube root, while [tex]\( \sqrt{5} \)[/tex] is a square root.
- Since [tex]\( \sqrt[3]{5} \)[/tex] and [tex]\( \sqrt{5} \)[/tex] are not the same type of root, [tex]\( x \times \sqrt[3]{5} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 2: [tex]\( \sqrt{5y} \)[/tex]
- This can be rewritten as [tex]\( \sqrt{5} \times \sqrt{y} \)[/tex].
- While this expression contains [tex]\( \sqrt{5} \)[/tex], it also involves an additional factor of [tex]\( \sqrt{y} \)[/tex].
- Because the radicand [tex]\( 5y \)[/tex] is different from [tex]\( 5 \)[/tex], [tex]\( \sqrt{5y} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 3: [tex]\( 3 \times \sqrt[3]{5x} \)[/tex]
- This option involves the cube root of [tex]\( 5x \)[/tex]. The expression [tex]\(\sqrt[3]{5x}\)[/tex] is a cube root.
- Similar to option 1, since [tex]\(\sqrt[3]{5x} \)[/tex] and [tex]\( \sqrt{5} \)[/tex] are not the same type of root, [tex]\( 3 \times \sqrt[3]{5x} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 4: [tex]\( y \times \sqrt{5} \)[/tex]
- This option involves [tex]\( y \)[/tex] as a scalar multiple of [tex]\( \sqrt{5} \)[/tex].
- The expression [tex]\( y \times \sqrt{5} \)[/tex] has the same radicand ([tex]\( 5 \)[/tex]) and the same type of root (square root) as [tex]\( 3 \times \sqrt{5} \)[/tex].
- Here, the radicand and the type of root match exactly with [tex]\( 3 \times \sqrt{5} \)[/tex], making this a like radical expression.
Thus, the correct option is:
[tex]\[ \boxed{y \times \sqrt{5}} \][/tex]
Let's analyze each option step by step:
### Option 1: [tex]\( x \times \sqrt[3]{5} \)[/tex]
- This option involves the cube root of [tex]\( 5 \)[/tex]. The expression [tex]\(\sqrt[3]{5}\)[/tex] is a cube root, while [tex]\( \sqrt{5} \)[/tex] is a square root.
- Since [tex]\( \sqrt[3]{5} \)[/tex] and [tex]\( \sqrt{5} \)[/tex] are not the same type of root, [tex]\( x \times \sqrt[3]{5} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 2: [tex]\( \sqrt{5y} \)[/tex]
- This can be rewritten as [tex]\( \sqrt{5} \times \sqrt{y} \)[/tex].
- While this expression contains [tex]\( \sqrt{5} \)[/tex], it also involves an additional factor of [tex]\( \sqrt{y} \)[/tex].
- Because the radicand [tex]\( 5y \)[/tex] is different from [tex]\( 5 \)[/tex], [tex]\( \sqrt{5y} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 3: [tex]\( 3 \times \sqrt[3]{5x} \)[/tex]
- This option involves the cube root of [tex]\( 5x \)[/tex]. The expression [tex]\(\sqrt[3]{5x}\)[/tex] is a cube root.
- Similar to option 1, since [tex]\(\sqrt[3]{5x} \)[/tex] and [tex]\( \sqrt{5} \)[/tex] are not the same type of root, [tex]\( 3 \times \sqrt[3]{5x} \)[/tex] is not like [tex]\( 3 \times \sqrt{5} \)[/tex].
### Option 4: [tex]\( y \times \sqrt{5} \)[/tex]
- This option involves [tex]\( y \)[/tex] as a scalar multiple of [tex]\( \sqrt{5} \)[/tex].
- The expression [tex]\( y \times \sqrt{5} \)[/tex] has the same radicand ([tex]\( 5 \)[/tex]) and the same type of root (square root) as [tex]\( 3 \times \sqrt{5} \)[/tex].
- Here, the radicand and the type of root match exactly with [tex]\( 3 \times \sqrt{5} \)[/tex], making this a like radical expression.
Thus, the correct option is:
[tex]\[ \boxed{y \times \sqrt{5}} \][/tex]
