What is [tex]\sqrt[4]{7^3}[/tex] in exponential form?

A. [tex]7^{\frac{3}{4}}[/tex]

B. [tex]7^{\frac{4}{3}}[/tex]

C. [tex]7^{-\frac{4}{3}}[/tex]



Answer :

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].