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

Aug 30, 2015

$x = 256$

#### Explanation:

$3 \cdot \sqrt[4]{{x}^{3}} = 192$

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

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot \sqrt[4]{{x}^{3}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} = \frac{192}{3}$

$\sqrt[4]{{x}^{3}} = 64$

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

${\left(\sqrt[4]{{x}^{3}}\right)}^{4} = {64}^{4}$

${x}^{3} = 64 \cdot {64}^{3}$

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

$\sqrt[3]{{x}^{3}} = \sqrt[3]{64 \cdot {64}^{3}}$

$x = \sqrt[3]{64} \cdot 64$

$x = 4 \cdot 64 = \textcolor{g r e e n}{256}$