How to solve Fraction Exponents

Exponential Equations with a radical in the exponent

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

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 1

Isolate fraction exponent

Step 2

Raise both sides to multiplicative inverse of the exponent

$$\left( x ^{\frac 2 3} \right)^{\red{\frac 3 2}} = 64^{\red{\frac 3 2}}$$
Step 2
$$x = 64^{\red{\frac 3 2}} \\ x = 512$$