Alright, let's break down the question step-by-step to find the values of each term individually and then sum them up.
We are given the expression:
[tex]\[
\left(\frac{3}{2}\right)^{-2} + \left(\frac{25}{81}\right)^{0.5} + \left(\frac{1}{16}\right)^{-1/4}
\][/tex]
### Step 1: Compute \(\left(\frac{3}{2}\right)^{-2}\)
First, let's deal with the term \(\left(\frac{3}{2}\right)^{-2}\):
[tex]\[
\left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^{2} = \frac{4}{9}
\][/tex]
So, \(\left(\frac{3}{2}\right)^{-2} = 0.4444444444444444\)
### Step 2: Compute \(\left(\frac{25}{81}\right)^{0.5}\)
Next, let's calculate \(\left(\frac{25}{81}\right)^{0.5}\):
[tex]\[
\left(\frac{25}{81}\right)^{0.5} = \sqrt{\frac{25}{81}} = \frac{\sqrt{25}}{\sqrt{81}} = \frac{5}{9}
\][/tex]
So, \(\left(\frac{25}{81}\right)^{0.5} = 0.5555555555555556\)
### Step 3: Compute \(\left(\frac{1}{16}\right)^{-1/4}\)
Now, we evaluate \(\left(\frac{1}{16}\right)^{-1/4}\):
A quick way is to remember that a negative exponent means a reciprocal, and a fractional exponent represents a root. So we can break this down:
[tex]\[
\left(\frac{1}{16}\right)^{-1/4} = \left(16\right)^{1/4} = \sqrt[4]{16} = 2
\][/tex]
So, \(\left(\frac{1}{16}\right)^{-1/4} = 2.0\)
### Step 4: Sum the terms
Finally, we sum up all the individual results:
[tex]\[
0.4444444444444444 + 0.5555555555555556 + 2.0 = 3.0
\][/tex]
Thus, the complete value of the expression:
[tex]\[
\left(\frac{3}{2}\right)^{-2} + \left(\frac{25}{81}\right)^{0.5} + \left(\frac{1}{16}\right)^{-1/4} = 3.0
\][/tex]
