To solve the expression [tex]\(\left(\frac{6}{5}\right)^{-8}\)[/tex], we need to understand the concept of negative exponents. Here's the detailed, step-by-step solution:
### Step 1: Understand the Negative Exponent
A negative exponent indicates that we take the reciprocal (or multiplicative inverse) of the base and then raise it to the positive exponent. In this case, [tex]\(\left(\frac{6}{5}\right)^{-8} \)[/tex] implies:
[tex]\[
\left(\frac{6}{5}\right)^{-8} = \left(\frac{1}{\frac{6}{5}}\right)^{8}
\][/tex]
### Step 2: Calculate the Reciprocal
To find the reciprocal of [tex]\(\frac{6}{5}\)[/tex], we simply invert the fraction:
[tex]\[
\frac{1}{\frac{6}{5}} = \frac{5}{6}
\][/tex]
### Step 3: Apply the Positive Exponent
Next, we raise the reciprocal [tex]\(\frac{5}{6} \)[/tex] to the power of 8:
[tex]\[
\left(\frac{5}{6}\right)^{8}
\][/tex]
### Step 4: Calculate the Result
The precise calculation of [tex]\(\left(\frac{5}{6}\right)^{8} \)[/tex] is:
[tex]\[
\left(\frac{5}{6}\right)^{8} = 0.2325680393613779
\][/tex]
So,
[tex]\[
\left(\frac{6}{5}\right)^{-8} = 0.2325680393613779
\][/tex]
Therefore, the value of [tex]\(\left(\frac{6}{5}\right)^{-8} \)[/tex] is approximately [tex]\(0.2325680393613779\)[/tex].
