Question 1209888: Compute
\frac{\{\sqrt{3}\} - 4 \{\sqrt{5}\}}{\{\sqrt{3}\}^2 + \{\sqrt{2}\}^2}.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's compute the given expression step-by-step.
**Understanding Fractional Parts**
The fractional part of a number $x$, denoted by $\{x\}$, is defined as $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
**Calculating Fractional Parts**
1. **$\{\sqrt{3}\}$:**
* $\sqrt{3} \approx 1.732$
* $\lfloor \sqrt{3} \rfloor = 1$
* $\{\sqrt{3}\} = \sqrt{3} - 1$
2. **$\{\sqrt{5}\}$:**
* $\sqrt{5} \approx 2.236$
* $\lfloor \sqrt{5} \rfloor = 2$
* $\{\sqrt{5}\} = \sqrt{5} - 2$
3. **$\{\sqrt{2}\}$:**
* $\sqrt{2} \approx 1.414$
* $\lfloor \sqrt{2} \rfloor = 1$
* $\{\sqrt{2}\} = \sqrt{2} - 1$
**Substituting into the Expression**
Now, substitute these fractional parts into the given expression:
$$\frac{\{\sqrt{3}\} - 4 \{\sqrt{5}\}}{\{\sqrt{3}\}^2 + \{\sqrt{2}\}^2} = \frac{(\sqrt{3} - 1) - 4(\sqrt{5} - 2)}{(\sqrt{3} - 1)^2 + (\sqrt{2} - 1)^2}$$
**Simplifying the Numerator**
* $(\sqrt{3} - 1) - 4(\sqrt{5} - 2) = \sqrt{3} - 1 - 4\sqrt{5} + 8 = \sqrt{3} - 4\sqrt{5} + 7$
**Simplifying the Denominator**
* $(\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2\sqrt{3} + 1 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$
* $(\sqrt{2} - 1)^2 = (\sqrt{2})^2 - 2\sqrt{2} + 1 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}$
* $(\sqrt{3} - 1)^2 + (\sqrt{2} - 1)^2 = (4 - 2\sqrt{3}) + (3 - 2\sqrt{2}) = 7 - 2\sqrt{3} - 2\sqrt{2}$
**Putting it Together**
The expression becomes:
$$\frac{\sqrt{3} - 4\sqrt{5} + 7}{7 - 2\sqrt{3} - 2\sqrt{2}}$$
**Approximating the Result**
Let's approximate the values:
* $\sqrt{3} \approx 1.732$
* $\sqrt{5} \approx 2.236$
* $\sqrt{2} \approx 1.414$
Numerator:
* $1.732 - 4(2.236) + 7 = 1.732 - 8.944 + 7 = -0.212$
Denominator:
* $7 - 2(1.732) - 2(1.414) = 7 - 3.464 - 2.828 = 0.708$
Then the fraction is:
$$\frac{-0.212}{0.708} \approx -0.299435$$
**Exact Result**
$$\frac{\sqrt{3} - 4\sqrt{5} + 7}{7 - 2\sqrt{3} - 2\sqrt{2}}$$
Final Answer: The final answer is $\boxed{\frac{7 + \sqrt{3} - 4 \sqrt{5}}{7 - 2 \sqrt{2} - 2 \sqrt{3}}}$
|
|
|
