Let's solve this step-by-step to find which of the given radical expressions is equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
First, we note that [tex]\( 7 \sqrt{3} \)[/tex] is a simplified form of a radical, and we want to determine which of the options given matches this.
### Option a: [tex]\( \sqrt{147} \)[/tex]
To check if [tex]\( \sqrt{147} \)[/tex] is equivalent to [tex]\( 7 \sqrt{3} \)[/tex], let's see if 147 can be broken down into factors that include 3.
[tex]\[
147 = 3 \times 49 = 3 \times 7^2
\][/tex]
Thus,
[tex]\[
\sqrt{147} = \sqrt{3 \times 7^2} = \sqrt{3} \times \sqrt{7^2} = \sqrt{3} \times 7 = 7 \sqrt{3}
\][/tex]
So, we see that:
[tex]\[
\sqrt{147} = 7 \sqrt{3}
\][/tex]
Therefore, option a is equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
### Option b: [tex]\( \sqrt{42} \)[/tex]
Next, we check if [tex]\( \sqrt{42} \)[/tex] can be simplified to match [tex]\( 7 \sqrt{3} \)[/tex].
[tex]\[
42 = 2 \times 21 = 2 \times 3 \times 7
\][/tex]
Thus,
[tex]\[
\sqrt{42} = \sqrt{2 \times 3 \times 7}
\][/tex]
This does not simplify to [tex]\( 7 \sqrt{3} \)[/tex], hence:
[tex]\[
\sqrt{42} \neq 7 \sqrt{3}
\][/tex]
So, option b is not equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
### Option c: [tex]\( \sqrt{63} \)[/tex]
For [tex]\( \sqrt{63} \)[/tex]:
[tex]\[
63 = 3 \times 21 = 3 \times 3 \times 7 = 3^2 \times 7
\][/tex]
Thus,
[tex]\[
\sqrt{63} = \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} = 3 \sqrt{7}
\][/tex]
This does not simplify to [tex]\( 7 \sqrt{3} \)[/tex], so:
[tex]\[
\sqrt{63} \neq 7 \sqrt{3}
\][/tex]
Therefore, option c is not equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
### Option d: [tex]\( \sqrt{21} \)[/tex]
Finally, let's check if [tex]\( \sqrt{21} \)[/tex] can be simplified to match [tex]\( 7 \sqrt{3} \)[/tex].
[tex]\[
21 = 3 \times 7
\][/tex]
Thus,
[tex]\[
\sqrt{21} = \sqrt{3 \times 7} = \sqrt{3} \times \sqrt{7}
\][/tex]
This does not simplify to [tex]\( 7 \sqrt{3} \)[/tex], hence:
[tex]\[
\sqrt{21} \neq 7 \sqrt{3}
\][/tex]
So, option d is not equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
### Conclusion
Only option a, [tex]\( \sqrt{147} \)[/tex], is equivalent to [tex]\( 7 \sqrt{3} \)[/tex].
[tex]\[
\boxed{\sqrt{147}}
\][/tex]
