Certainly! Let's go through the problem step-by-step to find the combined value of the given expression:
[tex]\[ \frac{1}{4} \sqrt{2} + \frac{3}{4} \sqrt{5} - \frac{5}{6} \sqrt{2} \][/tex]
Step 1: Identify the terms with like radicals
- The terms [tex]\(\frac{1}{4} \sqrt{2}\)[/tex] and [tex]\(-\frac{5}{6} \sqrt{2}\)[/tex] both contain the radical [tex]\(\sqrt{2}\)[/tex].
- The term [tex]\(\frac{3}{4} \sqrt{5}\)[/tex] contains the radical [tex]\(\sqrt{5}\)[/tex] and can be considered separately.
Step 2: Combine the like terms
First, let's combine the terms involving [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \frac{1}{4} \sqrt{2} - \frac{5}{6} \sqrt{2} \][/tex]
To combine these, we need a common denominator. The denominators are 4 and 6, and the least common multiple of these is 12.
Convert [tex]\(\frac{1}{4}\)[/tex] to a fraction with a denominator of 12:
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
Convert [tex]\(\frac{5}{6}\)[/tex] to a fraction with a denominator of 12:
[tex]\[ \frac{5}{6} = \frac{10}{12} \][/tex]
Now, combine these with the common denominator:
[tex]\[ \frac{3}{12} \sqrt{2} - \frac{10}{12} \sqrt{2} = \left(\frac{3}{12} - \frac{10}{12}\right) \sqrt{2} = \frac{-7}{12} \sqrt{2} \][/tex]
So, the combined value of the terms involving [tex]\(\sqrt{2}\)[/tex] is:
[tex]\[ \frac{-7}{12} \sqrt{2} \approx -0.8249579113843055 \][/tex]
Step 3: Handle the term with [tex]\(\sqrt{5}\)[/tex]
The term [tex]\(\frac{3}{4} \sqrt{5}\)[/tex] remains as it is:
[tex]\[ \frac{3}{4} \sqrt{5} \approx 1.6770509831248424 \][/tex]
Step 4: Combine all the results
Finally, add the results together:
[tex]\[ \frac{-7}{12} \sqrt{2} + \frac{3}{4} \sqrt{5} \approx -0.8249579113843055 + 1.6770509831248424 \][/tex]
[tex]\[ \approx 0.8520930717405368 \][/tex]
Conclusion
So, the value of the expression:
[tex]\[ \frac{1}{4} \sqrt{2} + \frac{3}{4} \sqrt{5} - \frac{5}{6} \sqrt{2} \][/tex]
is approximately [tex]\( 0.8520930717405368 \)[/tex].
To summarize, the combined intermediate and final results are as follows:
- [tex]\(\frac{1}{4} \sqrt{2} \approx 0.3535533905932738\)[/tex]
- [tex]\(\frac{3}{4} \sqrt{5} \approx 1.6770509831248424\)[/tex]
- [tex]\(-\frac{5}{6} \sqrt{2} \approx -1.1785113019775793\)[/tex]
- Combined [tex]\(\sqrt{2}\)[/tex] terms: [tex]\(\frac{-7}{12} \sqrt{2} \approx -0.8249579113843055\)[/tex]
- Final result: [tex]\( 0.8520930717405368 \)[/tex]
Therefore, we have the final result.
