Equations with Rational Exponents
Steps for solving equations with rational exponents:
Example 1: Solve over the real numbers: \(\left(3x-5\right)^{\frac{3}{4}}-2=6 \).
Steps for solving equations with rational exponents:
- Isolate the expression with the rational exponent.
- Raise each side to the reciprocal power of the rational exponent.
- Solve the equation.
- Check for extraneous solutions.
Example 1: Solve over the real numbers: \(\left(3x-5\right)^{\frac{3}{4}}-2=6 \).
\(\begin{align}\left(3x-5\right)^{\frac{3}{4}}&=8\ & \ &\text{1) Isolate the expression with the rational exponent.}\\
\left(\left(3x-5\right)^{\frac{3}{4}}\right)^{\frac{4}{3}}&=8\strut^{\frac{4}{3}}\ & \ &\text{2) Raise each side to the reciprocal power of the rational exponent.}\\
3x-5&=16\ & \ &\text{3) use the Power of a Power property to simplify the left side and evaluate}\ 8\strut^{\frac{4}{3}}\ \text{as (}\left(\sqrt[3]{8}\right)^4\ \text{which is 16).}\\
3x&=21\\
x&=7\end{align}\)
\left(\left(3x-5\right)^{\frac{3}{4}}\right)^{\frac{4}{3}}&=8\strut^{\frac{4}{3}}\ & \ &\text{2) Raise each side to the reciprocal power of the rational exponent.}\\
3x-5&=16\ & \ &\text{3) use the Power of a Power property to simplify the left side and evaluate}\ 8\strut^{\frac{4}{3}}\ \text{as (}\left(\sqrt[3]{8}\right)^4\ \text{which is 16).}\\
3x&=21\\
x&=7\end{align}\)
Check:
\(\left(3\left(7\right)-5\right)^{\frac{3}{4}}-2=6\)
\(\left(16\right)^{\frac{3}{4}}-2=6\)
\(8-2=6 \) (evaluate \(16^{\frac{3}{4}}\) as \(\left(\sqrt[4]{16}\right)^3\) which is \(8\))
\(6=6\) ✓
The solution is \(x=7\).
Example 2: Solve over the real numbers: \(\left(2x+7\right)^{\frac{2}{3}}=9\).
\(\left(3\left(7\right)-5\right)^{\frac{3}{4}}-2=6\)
\(\left(16\right)^{\frac{3}{4}}-2=6\)
\(8-2=6 \) (evaluate \(16^{\frac{3}{4}}\) as \(\left(\sqrt[4]{16}\right)^3\) which is \(8\))
\(6=6\) ✓
The solution is \(x=7\).
Example 2: Solve over the real numbers: \(\left(2x+7\right)^{\frac{2}{3}}=9\).