Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Simplify the expression:

[tex]\[ \left(\frac{3}{2}\right)^{-2} + \left(\frac{25}{81}\right)^{0.5} + \left(\frac{1}{16}\right)^{-1/4} \][/tex]


Sagot :

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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.