Raise both sides by the power of three.

\(\displaystyle (\sqrt[3]{3-9x} )^3= 4^3\)

\(\displaystyle 3-9x = 64\)

Subtract three from both sides.

\(\displaystyle 3-9x-3 = 64-3\)

\(\displaystyle -9x=61\)

Divide both sides by negative nine.

\(\displaystyle \frac{-9x}{-9}=\frac{61}{-9}\)

The answer is:  \(\displaystyle -\frac{61}{9}\)

Subtract two from both sides.

\(\displaystyle \sqrt[4]{3-x} +2-2= 5-2\)

\(\displaystyle \sqrt[4]{3-x}=3\)

Raise both sides by the fourth power.  This will eliminate the radical.

\(\displaystyle (\sqrt[4]{3-x})^4=3^4\)

\(\displaystyle 3-x=81\)

Add \(\displaystyle x\) on both sides.

\(\displaystyle 3-x+x=81+x\)

\(\displaystyle 3=81+x\)

Subtract 81 on both sides.

\(\displaystyle 3-81=81+x-81\)

\(\displaystyle -78=x\)

The answer is:  \(\displaystyle -78\)

Square both sides to eliminate the radical.

\(\displaystyle \sqrt{6x+7} ^2=( 3^{-1})^2\)

\(\displaystyle 6x+7 = 3^{-2}\)

Convert the negative exponent into a fraction.

\(\displaystyle x^{-a} =\frac{1}{x^a}\)

\(\displaystyle 6x+7 =\frac{1}{9}\)

Multiply by nine on both sides to eliminate the fraction.

\(\displaystyle 9(6x+7) =9(\frac{1}{9})\)

\(\displaystyle 54x+63 = 1\)

Subtract 63 on both sides.

\(\displaystyle 54x+63 -63= 1-63\)

\(\displaystyle 54x= -62\)

Divide by 54 on both sides.

\(\displaystyle \frac{54x}{54}=\frac{ -62}{54}\)

Reduce the fraction.

The answer is:  \(\displaystyle -\frac{31}{27}\)

Square both sides to eliminate the radical.

\(\displaystyle (x\sqrt{5x}) ^2= 25^2\)

The equation becomes:

\(\displaystyle x^2\cdot 5x = 625\)

Divide by five on both sides and combine the product of x-variables.

\(\displaystyle x^3=\frac{ 625}{5}\)

\(\displaystyle x^3 = 125\)

Cube root both sides.

\(\displaystyle \sqrt[3]{x^3} = \sqrt[3]{125}\)

The answer is:  \(\displaystyle 5\)

Square both sides.

\(\displaystyle (\sqrt{8-x}+\sqrt{9-x} )^2=2^2\)

We will need to simplify the left side using the FOIL method.

\(\displaystyle (\sqrt{8-x}+\sqrt{9-x} )(\sqrt{8-x}+\sqrt{9-x} )\)

A radical multiplied by itself becomes what's inside the radical.

The expression becomes:

\(\displaystyle (8-x)+(\sqrt{8-x})(\sqrt{9-x})+(\sqrt{9-x})(\sqrt{8-x})+(9-x)\)

Combine like-terms and rewrite the equation.

\(\displaystyle -2x+2\sqrt{8-x}\sqrt{9-x}+17 = 4\)

Isolate the radical by adding \(\displaystyle 2x\) and subtract 17 on both sides.  Then divide by two on both sides.

\(\displaystyle 2\sqrt{8-x}\sqrt{9-x} =2x-13\)

\(\displaystyle \frac{2\sqrt{8-x}\sqrt{9-x} }{2}=\frac{2x-13}{2}\)

\(\displaystyle \sqrt{8-x}\sqrt{9-x} = x-\frac{13}{2}\)

The radicals can be combined as one whole using the FOIL method.

\(\displaystyle \sqrt{72-17x+x^2} = x-\frac{13}{2}\)

Square both sides.

\(\displaystyle (\sqrt{72-17x+x^2}) ^2= (x-\frac{13}{2})^2\)

\(\displaystyle 72-17x+x^2 = x^2-13x+\frac{169}{4}\)

The \(\displaystyle x^2\) terms will cancel by subtracting the term on both side, and we can solve for \(\displaystyle x\).

\(\displaystyle 72-17x = -13x+\frac{169}{4}\)

Add \(\displaystyle 17x\) on both sides.

\(\displaystyle 72 = 4x+\frac{169}{4}\)

Multiply by four on both sides to eliminate the fraction.

\(\displaystyle 4[72 = 4x+\frac{169}{4}]\)

\(\displaystyle 288 = 16x+169\)

Solve for x.

\(\displaystyle 288-169 = 16x+169-169\)

\(\displaystyle 119 = 16x\)

The answer is:  \(\displaystyle \frac{119}{16}\)

Divide by four on both sides.

\(\displaystyle \frac{4\sqrt{3x-3} }{4}= \frac{12}{4}\)

The equation becomes:  

\(\displaystyle \sqrt{3x-3} =3\)

Square both sides.

\(\displaystyle (\sqrt{3x-3}) ^2=3^2\)

\(\displaystyle 3x-3= 9\)

Add three on both sides.

\(\displaystyle 3x-3+3= 9+3\)

\(\displaystyle 3x =12\)

Divide by three on both sides.

\(\displaystyle \frac{3x}{3} =\frac{12}{3}\)

The answer is:  \(\displaystyle 4\)

Multiply by \(\displaystyle x\) on both sides.

\(\displaystyle \frac{\sqrt{3x}}{x} \cdot x= 6\cdot x\)

Simplify both sides.

\(\displaystyle \sqrt{3x}=6x\)

Square both sides to eliminate the radical.

\(\displaystyle (\sqrt{3x} )^2=(6x)^2\)

Simplify both sides.

\(\displaystyle 3x= 36x^2\)

Divide by \(\displaystyle 36x\) on both sides.

\(\displaystyle \frac{3x}{36x}= \frac{36x^2}{36x}\)

The answer is:  \(\displaystyle x=\frac{1}{12}\)

Divide by two on both sides.  This is similar to multiplying by one-half.

\(\displaystyle \frac{2\sqrt[3]{3x}}{2} = \frac{2}{3}\cdot \frac{1}{2}\)

Simplify both sides.

\(\displaystyle \sqrt[3]{3x} = \frac{1}{3}\)

Cube both sides to eliminate the radical.

\(\displaystyle (\sqrt[3]{3x} )^3= (\frac{1}{3})^3\)

\(\displaystyle 3x=\frac{1}{27}\)

Divide by three on both sides.  This is similar to multiplying by one-third on both sides.

\(\displaystyle \frac{3x}{3}=\frac{1}{27} \cdot \frac{1}{3}\)

The answer is:  \(\displaystyle \frac{1}{81}\)

Subtract 31 from both sides of the equation.

\(\displaystyle \sqrt{x-9}+31-31 = 35-31\)

\(\displaystyle \sqrt{x-9}=4\)

Square both sides.

\(\displaystyle (\sqrt{x-9})^2=4^2\)

Simplify both sides.

\(\displaystyle x-9 =16\)

Add nine on both sides.

\(\displaystyle x-9 +9=16+9\)

The answer is:  \(\displaystyle 25\)

To eliminate the radical, we will have to cube both sides.

\(\displaystyle (\sqrt[3]{3x-7})^3=4^3\)

The equation becomes:

\(\displaystyle 3x-7=64\)

Add seven on both sides.

\(\displaystyle 3x=71\)

Divide by three on both sides.

\(\displaystyle \frac{3x}{3}=\frac{71}{3}\)

The answer is:  \(\displaystyle \frac{71}{3}\)