To simplify the expression [tex]\(\sqrt{50} - \sqrt{98} + \sqrt{162}\)[/tex], we can express each term in its simplified form:
1. Simplify [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\][/tex]
2. Simplify [tex]\(\sqrt{98}\)[/tex]:
[tex]\[
\sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}
\][/tex]
3. Simplify [tex]\(\sqrt{162}\)[/tex]:
[tex]\[
\sqrt{162} = \sqrt{81 \cdot 2} = \sqrt{81} \cdot \sqrt{2} = 9\sqrt{2}
\][/tex]
Now, combining these simplified terms:
[tex]\[
\sqrt{50} - \sqrt{98} + \sqrt{162} = 5\sqrt{2} - 7\sqrt{2} + 9\sqrt{2}
\][/tex]
Next, combine the terms involving [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
(5 - 7 + 9)\sqrt{2} = 7\sqrt{2}
\][/tex]
Hence, the simplified form of the given expression is:
[tex]\[
\sqrt{50} - \sqrt{98} + \sqrt{162} = 7\sqrt{2}
\][/tex]
We can also confirm this numerically:
- Approximate value of [tex]\(5\sqrt{2} \approx 7.071\)[/tex]
- Approximate value of [tex]\(7\sqrt{2} \approx 9.899\)[/tex]
- Approximate value of [tex]\(9\sqrt{2} \approx 12.728\)[/tex]
Combining them:
[tex]\[
7.071 - 9.899 + 12.728 \approx 9.899
\][/tex]
The simplest answer among the options is:
[tex]\[
7\sqrt{2} \approx 9.899
\][/tex]
Thus, the correct simplified form is:
[tex]\[
\boxed{7\sqrt{2}}
\][/tex]
