Radical Equations with One Radical
Steps for solving radical equations with one radical:
Example 1: Solve over the real numbers.: \(\large{\frac{2}{3}}\sqrt[3]{2x+1}=4\).
Steps for solving radical equations with one radical:
- Isolate the radical term.
- Raise each side to the power of the index of the radical.
- Solve the equation.
- Check for extraneous solutions.
Example 1: Solve over the real numbers.: \(\large{\frac{2}{3}}\sqrt[3]{2x+1}=4\).
\(\begin{align}\sqrt[3]{2x+1}&=6\ & \ &\text{1) Multiply both sides by}\ {\frac{3}{2}}\ \text{to isolate the radical.}\\
\left(\sqrt[3]{2x+1}\right)^3&=6^3\ & \ &\text{2) Raise both sides to the third power.}\\
2x+1&=216\ & \ &\text{3) Evaluate}\ 6^3.\\
2x&=215\ & \ &\text{4) Subtrac.t}\\
x&=\large{\frac{215}{2}}\ & \ &\text{5) Divide.}\end{align}\)
\left(\sqrt[3]{2x+1}\right)^3&=6^3\ & \ &\text{2) Raise both sides to the third power.}\\
2x+1&=216\ & \ &\text{3) Evaluate}\ 6^3.\\
2x&=215\ & \ &\text{4) Subtrac.t}\\
x&=\large{\frac{215}{2}}\ & \ &\text{5) Divide.}\end{align}\)
Check:
\(\large{\frac{2}{3}}\sqrt[3]{2\left(\frac{215}{2}\right)+1}=4\)
\(\large{\frac{2}{3}}\sqrt[3]{215+1}=4\)
\(\large{\frac{2}{3}}\sqrt[3]{216}=4\)
\(\large{\frac{2}{3}}\left(6\right)=4\)
\(4=4\) ✓
The solution is \(x=\large{\frac{215}{2}}\)
Example 2: Solve over the real numbers: \(\sqrt{5x+1}=x-1\).