Answer :
To simplify the expression [tex]\(5^{\frac{1}{3}} \cdot 5^{\frac{3}{5}}\)[/tex], we can use the property of exponents that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In our case, both exponents are powers of [tex]\(5\)[/tex], so we can add the exponents together. Here's how you can do it step by step:
1. Identify the exponents:
- The first exponent is [tex]\(\frac{1}{3}\)[/tex]
- The second exponent is [tex]\(\frac{3}{5}\)[/tex]
2. Add the exponents:
- To add these fractions, you need a common denominator. The denominators are 3 and 5, so the least common denominator is 15.
- Convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with a denominator of 15:
[tex]\[
\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\][/tex]
- Convert [tex]\(\frac{3}{5}\)[/tex] to a fraction with a denominator of 15:
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
- Add the fractions:
[tex]\[
\frac{5}{15} + \frac{9}{15} = \frac{14}{15}
\][/tex]
3. Write the expression with the new exponent:
- The simplified form of the expression is [tex]\(5^{\frac{14}{15}}\)[/tex].
Therefore, the answer is [tex]\(5^{\frac{14}{15}}\)[/tex], which corresponds to the first option provided in the list.
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In our case, both exponents are powers of [tex]\(5\)[/tex], so we can add the exponents together. Here's how you can do it step by step:
1. Identify the exponents:
- The first exponent is [tex]\(\frac{1}{3}\)[/tex]
- The second exponent is [tex]\(\frac{3}{5}\)[/tex]
2. Add the exponents:
- To add these fractions, you need a common denominator. The denominators are 3 and 5, so the least common denominator is 15.
- Convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with a denominator of 15:
[tex]\[
\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\][/tex]
- Convert [tex]\(\frac{3}{5}\)[/tex] to a fraction with a denominator of 15:
[tex]\[
\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
\][/tex]
- Add the fractions:
[tex]\[
\frac{5}{15} + \frac{9}{15} = \frac{14}{15}
\][/tex]
3. Write the expression with the new exponent:
- The simplified form of the expression is [tex]\(5^{\frac{14}{15}}\)[/tex].
Therefore, the answer is [tex]\(5^{\frac{14}{15}}\)[/tex], which corresponds to the first option provided in the list.
