To solve the expression [tex]\(\sqrt{7} \times \sqrt[5]{7}\)[/tex], we need to understand the properties of exponents and roots.
1. Convert roots to exponents:
The square root of a number [tex]\(a\)[/tex] can be written as [tex]\(a^{1/2}\)[/tex]. Similarly, the fifth root of [tex]\(a\)[/tex] can be written as [tex]\(a^{1/5}\)[/tex].
Therefore:
[tex]\[
\sqrt{7} = 7^{1/2}
\][/tex]
and
[tex]\[
\sqrt[5]{7} = 7^{1/5}
\][/tex]
2. Combine the exponents:
When multiplying two terms with the same base, we add the exponents. Hence:
[tex]\[
\sqrt{7} \times \sqrt[5]{7} = 7^{1/2} \times 7^{1/5} = 7^{1/2 + 1/5}
\][/tex]
3. Find the common denominator and add the fractions:
The common denominator of 2 and 5 is 10. So, we convert the fractions:
[tex]\[
\frac{1}{2} = \frac{5}{10}
\][/tex]
and
[tex]\[
\frac{1}{5} = \frac{2}{10}
\][/tex]
Adding these fractions gives:
[tex]\[
\frac{5}{10} + \frac{2}{10} = \frac{7}{10}
\][/tex]
4. Simplify the exponent:
So, the combined exponent is:
[tex]\[
7^{1/2 + 1/5} = 7^{7/10}
\][/tex]
5. Compute the value:
Knowing from the given information:
[tex]\[
7^{7/10} \approx 3.9045287771227226
\][/tex]
We now check which option is closest to this value.
6. Compare with given options:
A) [tex]\(\frac{71}{10} = 7.1\)[/tex] [tex]\[ \text{(Not a close match)} \][/tex]
B) 76 [tex]\[ \text{(Not a close match)} \][/tex]
C) [tex]\(\frac{77}{10} = 7.7\)[/tex] [tex]\[ \text{(Not a close match)} \][/tex]
D) [tex]\(\frac{1}{710} \approx 0.0014\)[/tex] [tex]\[ \text{(Not a close match)} \][/tex]
Given these comparisons, it seems there is no exact choice that matches the calculated value. However, the closest computed value of [tex]\(7^{7/10}\)[/tex] does not directly correspond to any of the given multiple-choice options.
