How do you solve and check for extraneous solutions in #3x^(3/4)=192#?

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Answer 1 · 2015-08-30T00:33:13

#x = 256#

First, rewrite your equation into radical form

#3 * root(4)(x^3) = 192#

Divide both sides to get the radical term alone on the left-hand side

#(color(red)(cancel(color(black)(3))) * root(4)(x^3))/color(red)(cancel(color(black)(3))) = 192/3#

#root(4)(x^3) = 64#

Next, raise both sides of the equation to the fourth power

#(root(4)(x^3))^4 = 64^4#

#x^3 = 64 * 64^3#

Now take the cube root of both sides of the equation to get - don't forget that when you take the odd root of a number you only get one solution.

This also implies that you won't have extraneous solutions, since you will be left with a positive number under an even root for the main equation

#root(3)(x^3) = root(3)(64 * 64^3)#

#x = root(3)(64) * 64#

#x = 4 * 64 = color(green)(256)#