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Select the correct answer.

Simplify the following expression: [tex]x^{\frac{1}{3}} \cdot x^{\frac{1}{5}}[/tex]

A. [tex]x^{\frac{1}{16}}[/tex]
B. [tex]x^{15}[/tex]
C. [tex]x^{\frac{8}{16}}[/tex]
D. [tex]x^{\frac{2}{15}}[/tex]


Sagot :

To simplify the expression [tex]\(x^{\frac{1}{3}} \cdot x^{\frac{1}{5}}\)[/tex], we can use the properties of exponents. Specifically, we use the property that when multiplying two exponents with the same base, we add the exponents together. Here's a step-by-step explanation:

1. Identify the exponents: We are multiplying [tex]\(x\)[/tex] raised to [tex]\(\frac{1}{3}\)[/tex] and [tex]\(x\)[/tex] raised to [tex]\(\frac{1}{5}\)[/tex].

2. Add the exponents:
[tex]\[ \frac{1}{3} + \frac{1}{5} \][/tex]

3. Find a common denominator to add the fractions:
- The common denominator for 3 and 5 is 15.

4. Convert the fractions:
[tex]\[ \frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{1}{5} = \frac{3}{15} \][/tex]

5. Add the converted fractions:
[tex]\[ \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \][/tex]

6. Combine the fractions:
[tex]\[ \frac{1}{3} + \frac{1}{5} = \frac{8}{15} \][/tex]

7. Simplified expression:
[tex]\[ x^{\frac{1}{3}} \cdot x^{\frac{1}{5}} = x^{\frac{8}{15}} \][/tex]

Based on the given choices:

A. [tex]\(\quad x^{\frac{1}{16}}\)[/tex]
B. [tex]\(\quad x^{15}\)[/tex]
C. [tex]\(\quad x^{\frac{8}{16}}\)[/tex]
D. [tex]\(\quad x^{\frac{2}{15}}\)[/tex]

The correct answer is:

[tex]\[ \boxed{x^{\frac{8}{15}}} \][/tex]