To express the radical form [tex]\(\sqrt[4]{7^3}\)[/tex] in exponential form, we need to understand how to convert from radical notation to exponential notation.
Recall that the [tex]\( n \)[/tex]-th root of a number [tex]\( a \)[/tex] can be written in exponential form as [tex]\( a^{\frac{1}{n}} \)[/tex]. In this specific example, we have the 4th root of [tex]\( 7^3 \)[/tex].
1. The 4th root of a number [tex]\( x \)[/tex] can be written as [tex]\( x^{\frac{1}{4}} \)[/tex].
2. Therefore, the 4th root of [tex]\( 7^3 \)[/tex] can be written as [tex]\( (7^3)^{\frac{1}{4}} \)[/tex].
Next, we use the property of exponents which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
3. Applying this property, we have:
[tex]\[
(7^3)^{\frac{1}{4}} = 7^{3 \cdot \frac{1}{4}} = 7^{\frac{3}{4}}
\][/tex]
Therefore, the correct exponential form of [tex]\(\sqrt[4]{7^3}\)[/tex] is [tex]\( 7^{\frac{3}{4}} \)[/tex].
To verify our result, let's consider the numerical value of [tex]\( 7^{\frac{3}{4}} \)[/tex].
4. [tex]\( 7^{\frac{3}{4}} \)[/tex] is approximately [tex]\( 4.303517070658851 \)[/tex].
This value also confirms our calculation, as it matches the numerical result provided.
Thus, the exponential form of [tex]\(\sqrt[4]{7^3}\)[/tex] is:
[tex]\[
7^{\frac{3}{4}}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{7^{\frac{3}{4}}}
\][/tex]
