When determining the correct radical expression corresponding to the given algebraic expression [tex]\(2d^1\)[/tex], let's carefully evaluate each provided option.
Given the expression is [tex]\(2d^1\)[/tex], we want to match this with one of the radical forms listed.
We have four options to consider:
1. [tex]\(\sqrt[7]{2 d^{10}}\)[/tex]
2. [tex]\(\sqrt[10]{2 d^7}\)[/tex]
3. [tex]\(2 \sqrt[10]{d^7}\)[/tex]
4. [tex]\(2 \sqrt[7]{d^{10}}\)[/tex]
Now, let's analyze each option step-by-step:
### Option 1: [tex]\(\sqrt[7]{2 d^{10}}\)[/tex]
This expression represents the seventh root of [tex]\(2d^{10}\)[/tex]. In simplified form, it would be difficult to equate this to [tex]\(2d^1\)[/tex] since neither the base numbers nor the powers directly correlate to a simplified linear form of [tex]\(2d\)[/tex].
### Option 2: [tex]\(\sqrt[10]{2 d^7}\)[/tex]
This expression denotes the tenth root of [tex]\(2d^7\)[/tex]. Simplified, it wouldn’t easily reduce to the given form [tex]\(2d^1\)[/tex], making it an improbable match.
### Option 3: [tex]\(2 \sqrt[10]{d^7}\)[/tex]
In this option, we have twice the [tex]\(10\)[/tex]th root of [tex]\(d^7\)[/tex]. Examining this, it simplifies as the constant 2 remains separate from the radical, suggesting that it can interact s a separate factor. This might be representative of our given form [tex]\(2d\)[/tex].
### Option 4: [tex]\(2 \sqrt[7]{d^{10}}\)[/tex]
This represents twice the seventh root of [tex]\(d^{10}\)[/tex]. Upon deeper evaluation, the power of [tex]\(d\)[/tex] and the root degree further complicate matching directly to [tex]\(2d\)[/tex].
Considering a streamlined conversion, observing that among the options, [tex]\(2 \sqrt[10]{d^7}\)[/tex] separates the constant [tex]\(2\)[/tex] which is preserved and interprets a possible transformation relevant to [tex]\(d\)[/tex] when simplified.
Based on the evaluation, the correct radical expression that best represents [tex]\(2d^1\)[/tex] is:
[tex]\(2 \sqrt[10]{d^7}\)[/tex]
So, the correct answer is:
[tex]\(2 \sqrt[10]{d^7}\)[/tex]
