Sure! Let's solve the given expression step by step:
The expression we need to solve is:
[tex]\[
\left(\frac{3}{2}\right)^{-2} + \left(\frac{25}{81}\right)^{0.5} + \left(\frac{1}{16}\right)^{-1/4}
\][/tex]
1. Calculate the first term:
[tex]\[
\left(\frac{3}{2}\right)^{-2}
\][/tex]
To deal with the negative exponent, we can rewrite this as the reciprocal:
[tex]\[
\left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}
\][/tex]
[tex]\[
\left(\frac{4}{9}\right) \approx 0.4444444444444444
\][/tex]
2. Calculate the second term:
[tex]\[
\left(\frac{25}{81}\right)^{0.5}
\][/tex]
The exponent \(0.5\) is equivalent to taking the square root:
[tex]\[
\left(\frac{25}{81}\right)^{0.5} = \sqrt{\frac{25}{81}} = \frac{\sqrt{25}}{\sqrt{81}} = \frac{5}{9}
\][/tex]
[tex]\[
\left(\frac{5}{9}\right) \approx 0.5555555555555556
\][/tex]
3. Calculate the third term:
[tex]\[
\left(\frac{1}{16}\right)^{-1/4}
\][/tex]
A negative exponent means taking the reciprocal, and \(1/4\) is the fourth root:
[tex]\[
\left(\frac{1}{16}\right)^{-1/4} = \left(\frac{16}{1}\right)^{1/4} = 16^{1/4}
\][/tex]
The fourth root of 16 is:
[tex]\[
16^{1/4} = 2
\][/tex]
4. Sum the terms:
Now add all of the terms together:
[tex]\[
0.4444444444444444 + 0.5555555555555556 + 2 = 3
\][/tex]
So, the value of the given expression is:
[tex]\[
\boxed{3}
\][/tex]
