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:  \(\displaystyle \frac{71}{3}\)

Divide by three on both sides.

\(\displaystyle \frac{3\sqrt[4]{9-2x} }{3}= \frac{1}{3}\)

Rewrite the equation.

\(\displaystyle \sqrt[4]{9-2x} = \frac{1}{3}\)

Raise both sides to the fourth power.

\(\displaystyle (\sqrt[4]{9-2x})^4 =( \frac{1}{3})^4\)

Simplify both sides.

\(\displaystyle 9-2x = \frac{1}{81}\)

Multiply by 81 on both sides to eliminate the fraction.

\(\displaystyle 81(9-2x )= \frac{1}{81} \cdot 81\)

\(\displaystyle 729-162x=1\)

Subtract 729 from both sides.

\(\displaystyle 729-162x-729=1-729\)

\(\displaystyle -162x = -728\)

Divide by negative 162 on both sides.

\(\displaystyle \frac{-162x}{-162} =\frac{ -728}{-162}\)

Reduce the fractions.  Double negatives will negate, turning the answer positive.

\(\displaystyle \frac{ -728}{-162} =\frac{364\times 2}{81\times 2}\)

The answer is:  \(\displaystyle \frac{364}{81}\)

Isolate the radical.  Add four on both sides.

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

\(\displaystyle 3\sqrt[4]{4-2x}=6\)

Divide by three on both sides.  The equation becomes:

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

Raise both sides by four.

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

\(\displaystyle 4-2x = 16\)

Subtract four on both sides.

\(\displaystyle 4-2x -4= 16-4\)

\(\displaystyle -2x=12\)

Divide by negative two on both sides.

\(\displaystyle \frac{-2x}{-2}=\frac{12}{-2}\)

The answer is:  \(\displaystyle -6\)

Multiply by six on both sides of the equation.

\(\displaystyle \frac{\sqrt[3]{5x-2}}{6} \cdot 6= \frac{1}{2} \cdot 6\)

The equation becomes:

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

Cube both sides to eliminate the radical.

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

\(\displaystyle 5x-2 = 27\)

Add two on both sides.

\(\displaystyle 5x-2 +2= 27+2\)

\(\displaystyle 5x=29\)

Divide by five on both sides.

\(\displaystyle \frac{5x}{5}=\frac{29}{5}\)

The answer is:  \(\displaystyle \frac{29}{5}\)

Add seven on both sides.

\(\displaystyle \sqrt{-6x-5}-7 +7= 10+7\)

\(\displaystyle \sqrt{-6x-5} = 17\)

Square both sides.  This will eliminate the radical.

\(\displaystyle (\sqrt{-6x-5})^2= 17^2\)

\(\displaystyle -6x-5 = 289\)

Add five on both sides.

\(\displaystyle -6x-5 +5= 289+5\)

\(\displaystyle -6x=294\)

Divide both sides by negative 6 and reduce the fraction.

\(\displaystyle \frac{-6x}{-6}=\frac{294}{-6}\)

The answer is:  \(\displaystyle -49\)

Cube both sides of the equation.

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

This will eliminate the radical on the left side.

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

\(\displaystyle 3x= \frac{8}{27}\)

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

\(\displaystyle 3x \times \frac{1}{3}= \frac{8}{27} \times \frac{1}{3}=\frac{8}{81}\)

The answer is:  \(\displaystyle \frac{8}{81}\)

Subtract both sides by three.

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

The equation becomes: 

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

Divide both sides by negative one.

\(\displaystyle \frac{-\sqrt[3]{2-x} }{-1}=\frac{-4}{-1}\)

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

Cube both sides to eliminate the radical.

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

\(\displaystyle 2-x = 64\)

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

\(\displaystyle 2-x +x= 64+x\)

\(\displaystyle 2= 64+x\)

Subtract 64 on both sides.

\(\displaystyle 2-64= 64+x-64\)

The answer is:  \(\displaystyle -62\)

Raise both sides by the power of eight to eliminate the radical.

\(\displaystyle (\sqrt[8]{2x-1} )^8=2^8\)

Simplify both sides.

\(\displaystyle 2x-1=256\)

Add one on both sides.

\(\displaystyle 2x-1+1=256+1\)

\(\displaystyle 2x=257\)

Divide by two on both sides.

\(\displaystyle \frac{2x}{2}=\frac{257}{2}\)

The answer is:  \(\displaystyle \frac{257}{2}\)

Multiply by \(\displaystyle \sqrt{3x}\) on both sides.

\(\displaystyle \frac{1}{\sqrt{3x}} \cdot \sqrt{3x}= 4 \cdot \sqrt{3x}\)

\(\displaystyle 1=4\sqrt{3x}\)

Divide by 4 on both sides.

\(\displaystyle \frac{1}{4}=\frac{4\sqrt{3x}}{4}\)

\(\displaystyle \frac{1}{4} =\sqrt{3x}\)

Square both sides to eliminate the radical.

\(\displaystyle (\frac{1}{4}) ^2=(\sqrt{3x})^2\)

\(\displaystyle \frac{1}{16} = 3x\)

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

\(\displaystyle \frac{1}{16} \cdot \frac{1}{3}= 3x\cdot \frac{1}{3}\)

The answer is:  \(\displaystyle \frac{1}{48}\)