To solve the expression [tex]\(\sqrt[4]{7^3}\)[/tex] in exponential form step-by-step, we can use the properties of exponents and radicals.
1. Understand the notation: The fourth root of a number [tex]\(a\)[/tex] can be written as [tex]\(a^{\frac{1}{4}}\)[/tex]. Similarly, the cube of a number [tex]\(a\)[/tex] can be written as [tex]\(a^3\)[/tex].
2. Combine the exponents: The expression [tex]\(\sqrt[4]{7^3}\)[/tex] asks for the fourth root of [tex]\(7^3\)[/tex]. When dealing with roots and powers together, you can combine them into a single exponent by multiplying the exponents:
[tex]\[
\sqrt[4]{7^3} = (7^3)^{\frac{1}{4}}
\][/tex]
3. Apply the power rule: When you have a power raised to another power, you multiply the exponents:
[tex]\[
(7^3)^{\frac{1}{4}} = 7^{3 \cdot \frac{1}{4}}
\][/tex]
4. Simplify the exponent: Multiply the exponents together:
[tex]\[
7^{3 \cdot \frac{1}{4}} = 7^{\frac{3}{4}}
\][/tex]
Therefore, [tex]\(\sqrt[4]{7^3}\)[/tex] in exponential form is [tex]\(7^{\frac{3}{4}}\)[/tex].
Among the options given:
\- [tex]\(7^{\frac{3}{4}}\)[/tex]
\- [tex]\(7^{\frac{4}{3}}\)[/tex]
\- [tex]\(7^{-\frac{4}{3}}\)[/tex]
The correct answer is [tex]\(7^{\frac{3}{4}}\)[/tex].
