Sure, let's break down the given expression step-by-step to arrive at the solution.
The given expression is:
[tex]\[
\frac{\frac{6}{4}^{-2} \cdot \frac{6}{4} - 4 \cdot \frac{6}{4}}{\frac{6}{4}^{-2} \cdot \frac{6}{4} - 8 \cdot \left( -\frac{6}{4} \right)^2}
\][/tex]
First, let’s define the fraction [tex]$\frac{6}{4}$[/tex]:
[tex]\[
\frac{6}{4} = 1.5
\][/tex]
Now calculate each component separately:
1. Calculate [tex]\(\left( \frac{6}{4} \right)^{-2}\)[/tex]:
[tex]\[
\left( \frac{6}{4} \right)^{-2} = \left( 1.5 \right)^{-2} = \frac{1}{(1.5)^2} = \frac{1}{2.25} = 0.4444\ldots
\][/tex]
2. Calculate [tex]\(\frac{6}{4}^{-2} \cdot \frac{6}{4}\)[/tex]:
[tex]\[
0.4444\ldots \times 1.5 = 0.6667\ldots
\][/tex]
Here we round it to [tex]\(0.6667\)[/tex].
3. Calculate [tex]\(4 \cdot \frac{6}{4}\)[/tex]:
[tex]\[
4 \times 1.5 = 6.0
\][/tex]
4. Calculate the numerator [tex]\(\frac{6}{4}^{-2} \cdot \frac{6}{4} - 4 \cdot \frac{6}{4}\)[/tex]:
[tex]\[
0.6667 - 6.0 = -5.3333
\][/tex]
5. Calculate [tex]\(\left( -\frac{6}{4} \right)^{2}\)[/tex]:
[tex]\[
\left( -1.5 \right)^{2} = 2.25
\][/tex]
6. Calculate [tex]\(8 \cdot \left( -\frac{6}{4} \right)^2\)[/tex]:
[tex]\[
8 \times 2.25 = 18.0
\][/tex]
7. Combine terms for the denominator [tex]\(\frac{6}{4}^{-2} \cdot \frac{6}{4} - 8 \cdot \left( -\frac{6}{4} \right)^2\)[/tex]:
[tex]\[
0.6667 - 18.0 = -17.3333
\][/tex]
Finally, calculate the entire expression:
[tex]\[
\frac{-5.3333}{-17.3333} = 0.3077
\][/tex]
Thus, the final result of the expression is:
[tex]\[
\frac{\frac{6}{4}^{-2} \cdot \frac{6}{4} - 4 \cdot \frac{6}{4}}{\frac{6}{4}^{-2} \cdot \frac{6}{4} - 8 \cdot \left( -\frac{6}{4} \right)^2} = 0.3077
\][/tex]
