Certainly! Let's break down the given expressions step by step and find the required value of [tex]\(x\)[/tex].
### Step-by-Step Solution
1. Evaluate the Given Terms:
- [tex]\( \sqrt{5} \approx 2.2361 \)[/tex]
- [tex]\( \sqrt{10} \approx 3.1623 \)[/tex]
- [tex]\( \sqrt{15} \approx 3.8730 \)[/tex]
- [tex]\( \sqrt{6} \approx 2.4495 \)[/tex]
2. Calculate the First Expression:
- Let's simplify the expression: [tex]\( \sqrt{5} - \sqrt{10} - \sqrt{15} + \sqrt{6} \)[/tex]
- Substituting the values, we get:
[tex]\[
\sqrt{5} - \sqrt{10} - \sqrt{15} + \sqrt{6} \approx 2.2361 - 3.1623 - 3.8730 + 2.4495
\][/tex]
- Performing the arithmetic, we obtain:
[tex]\[
2.2361 - 3.1623 - 3.8730 + 2.4495 \approx -2.3497
\][/tex]
3. Check the Second Expression Equals [tex]\(\sqrt{x}\)[/tex]:
- According to the given condition:
[tex]\[
10 - \sqrt{15} + \sqrt{6} = \sqrt{x}
\][/tex]
- Substituting the values, we get:
[tex]\[
10 - 3.8730 + 2.4495 \approx 10 - 3.8730 + 2.4495 \approx 8.5765
\][/tex]
4. Square Both Sides to Find [tex]\(x\)[/tex]:
- Given that [tex]\( 10 - \sqrt{15} + \sqrt{6} = \sqrt{x} \)[/tex], we square both sides to find [tex]\(x\)[/tex]:
[tex]\[
(10 - \sqrt{15} + \sqrt{6})^2 = x
\][/tex]
- Calculating the right side:
[tex]\[
(8.5765)^2 \approx 73.5565
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is approximately [tex]\(73.5565\)[/tex].
