To determine which equations result in extraneous solutions, let's analyze each given equation.
1. [tex]\(\sqrt{x} = 5\)[/tex]
2. [tex]\(\sqrt{x} = -5\)[/tex]
3. [tex]\(\sqrt[3]{x} = 5\)[/tex]
4. [tex]\(\sqrt[3]{x} = -5\)[/tex]
5. [tex]\(\sqrt[4]{x-2} = -2\)[/tex]
6. [tex]\(\sqrt[4]{x+3} = 4\)[/tex]
7. [tex]\(\sqrt[4]{x+1} = -2\)[/tex]
8. [tex]\(\sqrt[7]{x+3} = -3\)[/tex]
### Equations Resulting in Extraneous Solutions:
- [tex]\(\sqrt{x} = -5\)[/tex]: This is an extraneous solution because the square root of a number cannot be negative.
- [tex]\(\sqrt[4]{x-2} = -2\)[/tex]: This is an extraneous solution because the fourth root of a number cannot be negative.
- [tex]\(\sqrt[4]{x+1} = -2\)[/tex]: Again, the fourth root of a number cannot be negative.
- [tex]\(\sqrt[7]{x+3} = -3\)[/tex]: This is an extraneous solution because although the seventh root can be negative, solving this equation results in [tex]\(x\)[/tex] such that [tex]\(x+3\)[/tex] yields a real number whose seventh root is indeed [tex]\(-3\)[/tex].
Therefore, the equations that will result in extraneous solutions are:
[tex]\[ \sqrt{x} = -5, \quad \sqrt[4]{x-2} = -2, \quad \sqrt[4]{x+1} = -2, \quad \sqrt[7]{x+3} = -3 \][/tex]
### Equations Without Extraneous Solutions:
- [tex]\(\sqrt{x} = 5\)[/tex]: The square root of [tex]\(x\)[/tex] can be 5.
- [tex]\(\sqrt[3]{x} = 5\)[/tex]: The cube root of [tex]\(x\)[/tex] can be 5.
- [tex]\(\sqrt[3]{x} = -5\)[/tex]: The cube root of [tex]\(x\)[/tex] can be -5.
- [tex]\(\sqrt[4]{x+3} = 4\)[/tex]: The fourth root of [tex]\(x+3\)[/tex] can be 4.
Therefore, the equations without extraneous solutions are:
[tex]\[ \sqrt{x} = 5, \quad \sqrt[3]{x} = 5, \quad \sqrt[3]{x} = -5, \quad \sqrt[4]{x+3} = 4 \][/tex]
### Final Table:
[tex]\[
\begin{array}{|l|l|}
\hline
\text{Extraneous Solutions} & \text{No Extraneous Solutions} \\
\hline
\sqrt{x} = -5 & \sqrt{x} = 5 \\
\sqrt[4]{x-2} = -2 & \sqrt[3]{x} = 5 \\
\sqrt[4]{x+1} = -2 & \sqrt[3]{x} = -5 \\
\sqrt[7]{x+3} = -3 & \sqrt[4]{x+3} = 4 \\
\hline
\end{array}
\][/tex]
