Radical Expressions

Question

How do you know what power to raise both sides of an equation to if the indexes of the radicals are different? For example, in √x+1=3^√x−1, what index would you use?

Answer 1

OK--this equation:

(x + 1)^1/2 = (x - 1)^1/3

^{I think the trick is to get to integer exponents. Try raising each side to the 6th power. This means multiply exponents:}

^{So we then have:}

(x + 1)³ = (x - 1)²

Expand:

x³ + 3x² + 3x + 1 = x² - 2x + 1

Collect terms:

x³ + 2x² + 5x = 0

Factor:

x (x2 + 2x + 5) = 0

There is only one real solution at x = 0.