Question 3 (Multiple Choice Worth 2 points)
(01.01 MC)

Which of the following is the simplified form of [tex] \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} [/tex]?

A. [tex] x^{\frac{1}{5}} [/tex]
B. [tex] x^{\frac{4}{5}} [/tex]
C. [tex] x^{\frac{4}{20}} [/tex]



Answer :

To simplify the expression [tex]\(\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x}\)[/tex], we need to use the rules of exponents.

1. Recall that the fifth root of [tex]\(x\)[/tex] can be written as an exponent:
[tex]\[ \sqrt[5]{x} = x^{\frac{1}{5}} \][/tex]

2. Given the expression, we now have:
[tex]\[ \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} = x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \][/tex]

3. When multiplying expressions with the same base, we add the exponents. Therefore:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot x^{\frac{1}{5}} = x^{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}} \][/tex]

4. Simplify the exponents by adding them together:
[tex]\[ \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5} \][/tex]

So, the simplified form of [tex]\(\sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x} \cdot \sqrt[5]{x}\)[/tex] is:
[tex]\[ x^{\frac{4}{5}} \][/tex]

Therefore, the correct answer is:
[tex]\[ x^{\frac{4}{5}} \][/tex]