Certainly! Let's solve the expression step-by-step.
Given the expression:
[tex]\[ \left[\frac{1}{\sqrt[5]{5}}\right]^{-10} \][/tex]
First, let's rewrite it in a clearer form:
[tex]\[ \left(\frac{1}{5^{1/5}}\right)^{-10} \][/tex]
Next, we'll recall the property of exponents that states:
[tex]\[ \left(a^m\right)^n = a^{mn} \][/tex]
In our case, [tex]\( a = \frac{1}{5^{1/5}} \)[/tex], [tex]\( m = 1 \)[/tex], and [tex]\( n = -10 \)[/tex]. Applying this property:
[tex]\[ \left(\frac{1}{5^{1/5}}\right)^{-10} = \left(5^{-1/5}\right)^{-10} \][/tex]
Now, we simplify the exponent by multiplying the exponents:
[tex]\[ 5^{-1/5 \times -10} = 5^{2} \][/tex]
Which simplifies to:
[tex]\[ 5^2 \][/tex]
Finally, calculate the value of [tex]\( 5^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
So, the result of the given expression is:
[tex]\[ \left[\frac{1}{\sqrt[5]{5}}\right]^{-10} = 25 \][/tex]
