Which of the following is equivalent to [tex](5)^{\frac{7}{3}}[/tex]?

A. [tex]5^{-4}[/tex]
B. [tex]5^4[/tex]
C. [tex]\sqrt[7]{5^3}[/tex]
D. [tex]\sqrt[3]{5^7}[/tex]



Answer :

To determine which expression is equivalent to [tex]\((5)^{\frac{7}{3}}\)[/tex], we need to recognize how fractional exponents work.

The expression [tex]\((5)^{\frac{7}{3}}\)[/tex] can be understood by breaking down the fraction in the exponent:
- The denominator (3) represents the root.
- The numerator (7) represents the power.

Thus, [tex]\((5)^{\frac{7}{3}}\)[/tex] is equivalent to taking the cube root of [tex]\(5\)[/tex] raised to the 7th power. Mathematically, this is expressed as:

[tex]\[ (5)^{\frac{7}{3}} = \sqrt[3]{5^7} \][/tex]

Among the given options:
1. [tex]\(5^{-4}\)[/tex] - This represents raising 5 to the power of -4. This is not equivalent to our expression.
2. [tex]\(5^4\)[/tex] - This represents raising 5 to the power of 4. This is also not equivalent to our expression.
3. [tex]\(\sqrt[7]{5^3}\)[/tex] - This represents the 7th root of [tex]\(5^3\)[/tex], which does not match our expression.
4. [tex]\(\sqrt[3]{5^7}\)[/tex] - This represents the cube root of [tex]\(5^7\)[/tex], which is exactly the same as our expression [tex]\((5)^{\frac{7}{3}}\)[/tex].

Therefore, the correct equivalent expression to [tex]\((5)^{\frac{7}{3}}\)[/tex] is:
[tex]\[ \sqrt[3]{5^7} \][/tex]