To begin solving this problem, find the greatest perfect square for all quantities under a radical. \(\displaystyle 3x\sqrt[4]{9072}\) might seem intimidating, but remember that raising even single-digit numbers to the fourth power creates huge numbers. In this case, \(\displaystyle 9072\) is divisible by \(\displaystyle 1296\), a perfect fourth power.

\(\displaystyle \frac{1}{2}x\sqrt[4]{9072} + x\sqrt[3]{54} = 18\) ---> \(\displaystyle \frac{1}{2}x\sqrt[4]{7\cdot 1296} + x\sqrt[3]{2\cdot 27} = 18\)

Pull the perfect terms out of each term on the left:

\(\displaystyle \frac{1}{2}x\sqrt[4]{7\cdot 1296} + x\sqrt[3]{2\cdot 27} = 18\) ---> \(\displaystyle 3x\sqrt[4]{7} + 3x\sqrt[3]{2} = 18\)

Next, factor out \(\displaystyle 3x\) from the left-hand side:

\(\displaystyle 3x\sqrt[4]{7} + 3x\sqrt[3]{2} = 18\) ---> \(\displaystyle 3x(\sqrt[4]{7} + \sqrt[3]{2}) = 18\)

Lastly, isolate \(\displaystyle x\), remembering to simplify the fraction where possible:

\(\displaystyle 3x(\sqrt[4]{7} + \sqrt[3]{2}) = 18\) ---> \(\displaystyle x = \frac{6}{(\sqrt[4]{7} + \sqrt[3]{2})}\)

To start, begin pulling the largest perfect square you can out of each number:

\(\displaystyle \sqrt{384} = \sqrt {4 \cdot 4 \cdot 4 \cdot 6 } = \sqrt {64 \cdot 6} = 8\sqrt 6\)

\(\displaystyle \sqrt {216} = \sqrt {6 \cdot 6 \cdot 6} = \sqrt {36 \cdot 6} = 6\sqrt 6\)

So, \(\displaystyle \sqrt{384} + \sqrt{216} = 8\sqrt 6 + 6\sqrt 6 = 14\sqrt{6}\). You can just add the two terms together once they have a common radical.

Again here, it is easiest to recognize that both of our terms are divisible by \(\displaystyle 7\), a prime number likely to appear in our final answer:

\(\displaystyle \sqrt{6300} - \sqrt{700} = \sqrt{7\cdot 900} - \sqrt{7 \cdot 100}\)

Now, simplify our perfect squares:

\(\displaystyle \sqrt{7\cdot 900} - \sqrt{7 \cdot 100} = 30\sqrt 7 - 10 \sqrt 7\)

Lastly, subtract our terms with a common radical:

\(\displaystyle 30\sqrt{7} - 10\sqrt {7} = 20\sqrt{7}\)

To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

\(\displaystyle x\sqrt{45} + x\sqrt{108} = 12\) ---> \(\displaystyle x\sqrt{5 \cdot 9} + x\sqrt{3\cdot 36 } = 12\)

Factor out the square root of each perfect square:

\(\displaystyle x\sqrt{5 \cdot 9} + x\sqrt{3 \cdot 36} = 12\) ---> \(\displaystyle 3x\sqrt{5} + 6x\sqrt{3} = 12\)

Next, factor out \(\displaystyle 3x\) from each term on the left-hand side of the equation:

\(\displaystyle 3x\sqrt{5} + 3x\sqrt{12} = 12\) ---> \(\displaystyle 3x(\sqrt{5} +2\sqrt{3}) = 12\)

Lastly, isolate \(\displaystyle x\):

\(\displaystyle 3x(\sqrt{5} +2\sqrt{3}) = 12\) ---> \(\displaystyle x = \frac{4}{(\sqrt{5} +2\sqrt{3})}\)