Sure, let's solve the expression:
[tex]\[ \left\lvert \frac{1}{4} \sqrt{2} + \frac{3}{4} \sqrt{5} - \frac{5}{6} \sqrt{2} \right\rvert \][/tex]
First, we need to calculate the value of each term separately.
1. Calculate [tex]\(\frac{1}{4} \sqrt{2}\)[/tex]:
[tex]\[
a = \frac{1}{4} \sqrt{2} \approx 0.3535533905932738
\][/tex]
2. Calculate [tex]\(\frac{3}{4} \sqrt{5}\)[/tex]:
[tex]\[
b = \frac{3}{4} \sqrt{5} \approx 1.6770509831248424
\][/tex]
3. Calculate [tex]\(\frac{5}{6} \sqrt{2}\)[/tex]:
[tex]\[
c = \frac{5}{6} \sqrt{2} \approx 1.1785113019775793
\][/tex]
Now, let's combine these values in the given expression:
[tex]\[ \frac{1}{4} \sqrt{2} + \frac{3}{4} \sqrt{5} - \frac{5}{6} \sqrt{2} \][/tex]
Substitute the calculated values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the expression:
[tex]\[ 0.3535533905932738 + 1.6770509831248424 - 1.1785113019775793 \][/tex]
Next, perform the addition and subtraction:
[tex]\[ (\approx 0.3535533905932738 + \approx 1.6770509831248424) \approx 2.0306043737181162 \][/tex]
[tex]\[ 2.0306043737181162 - 1.1785113019775793 \approx 0.8520930717405371 \][/tex]
Next, we need to take the absolute value of the result.
Since the value [tex]\(0.8520930717405371\)[/tex] is already positive, its absolute value is the same:
[tex]\[ \left\lvert \approx 0.8520930717405371 \right\rvert = 0.8520930717405371 \][/tex]
Thus, the final result of the expression is:
[tex]\[ 0.8520930717405371 \][/tex]
