Sure, let's walk through the solution step-by-step for the expression [tex]\(\left(\frac{1}{343}\right)^{\frac{2}{5}}\)[/tex].
1. Understanding the Expression:
- We are given a fraction raised to an exponent. Specifically, we have [tex]\(\frac{1}{343}\)[/tex] raised to the power of [tex]\(\frac{2}{5}\)[/tex].
2. Simplifying the Base:
- [tex]\(\frac{1}{343}\)[/tex] is a fraction where 343 is the denominator. Note that 343 can also be written as [tex]\(7^3\)[/tex]. So, [tex]\(\frac{1}{343} = \left(\frac{1}{7^3}\right)\)[/tex].
3. Rewriting using Exponents:
- Rewriting the fraction with an exponent: [tex]\(\frac{1}{7^3} = 7^{-3}\)[/tex].
4. Applying the Exponent:
- Now, we raise [tex]\(7^{-3}\)[/tex] to the power of [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
\left(7^{-3}\right)^{\frac{2}{5}}
\][/tex]
5. Multiplying the Exponents:
- When raising a power to another power, we multiply the exponents:
[tex]\[
7^{-3 \cdot \frac{2}{5}} = 7^{-\frac{6}{5}}
\][/tex]
6. Interpreting the Negative Exponent:
- A negative exponent means taking the reciprocal:
[tex]\[
7^{-\frac{6}{5}} = \frac{1}{7^{\frac{6}{5}}}
\][/tex]
7. Understanding [tex]\(7^{\frac{6}{5}}\)[/tex]:
- The exponent [tex]\(\frac{6}{5}\)[/tex] can be interpreted as taking the fifth root of 7 raised to the 6th power:
[tex]\[
7^{\frac{6}{5}} = (7^6)^{\frac{1}{5}}
\][/tex]
8. Combining the Results:
- Thus, [tex]\(\frac{1}{7^{\frac{6}{5}}}\)[/tex] is the final simplified form of [tex]\(\left(\frac{1}{343}\right)^{\frac{2}{5}}\)[/tex].
After evaluating the complete expression, the numerical result is:
[tex]\[
\left(\frac{1}{343}\right)^{\frac{2}{5}} = 0.09680155905721155
\][/tex]
This value is the final answer for the given expression.
