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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

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 1

Isolate fraction exponent

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

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