Before attemptng to solve rational exponential equations like $$ x ^{\frac{2}{3}} + 1 = 65 $$, you should be comfortable

- simplify fraction exponents like $$ 27^{\frac 2 3 } $$
- solve exponential equations such as $$ 9^x = 27^2 $$
- understand what a multiplicative inverse . ( lesson)

### How to Solve Equations with Rational Exponents

### Steps

- Isolate the expression with the fractional exponent
- Raise both sides to multiplicative inverse of the exponent
- simplify

##### Example 1

Solve $$ x ^{\frac{2}{3}} +1 = 65 $$

Step 1Isolate fraction exponent

Raise both sides to multiplicative inverse of the exponent

$$
\left( x ^{\frac 2 3} \right)^{\red{\frac 3 2}} = 64^{\red{\frac 3 2}}
$$

$$
x = 64^{\red{\frac 3 2}}
\\
x = 512
$$