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Algebra II : Solving Radical Equations

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Example Questions

Example Question #71 : Solving Radical Equations

Solve the equation: $\sqrt[3]{3x} = \frac{2}{3}$

Possible Answers:

$\frac{16}{81}$

$\frac{8}{9}$

$\frac{1}{3}$

$\frac{2}{27}$

$\frac{8}{81}$

Correct answer:

$\frac{8}{81}$

Explanation:

Cube both sides of the equation.

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

This will eliminate the radical on the left side.

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

$3x= \frac{8}{27}$

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

$3x \times \frac{1}{3}= \frac{8}{27} \times \frac{1}{3}=\frac{8}{81}$

The answer is: $\frac{8}{81}$

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Example Question #72 : Solving And Graphing Radicals

Solve the radical equation: $3-\sqrt[3]{2-x} = -1$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

Subtract both sides by three.

$3-\sqrt[3]{2-x} -3= -1-3$

The equation becomes:

$-\sqrt[3]{2-x} =-4$

Divide both sides by negative one.

$\frac{-\sqrt[3]{2-x} }{-1}=\frac{-4}{-1}$

$\sqrt[3]{2-x} =4$

Cube both sides to eliminate the radical.

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

2-x = 64

Add on both sides.

2-x +x= 64+x

2= 64+x

Subtract 64 on both sides.

2-64= 64+x-64

The answer is:

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Example Question #73 : Solving And Graphing Radicals

Solve the radical equation: $\sqrt[8]{2x-1} =2$

Possible Answers:

$\frac{255}{2}$

$\frac{63}{2}$

$\frac{17}{2}$

$\frac{65}{2}$

$\frac{257}{2}$

Correct answer:

$\frac{257}{2}$

Explanation:

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

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

Simplify both sides.

2x-1=256

Add one on both sides.

2x-1+1=256+1

2x=257

Divide by two on both sides.

$\frac{2x}{2}=\frac{257}{2}$

The answer is: $\frac{257}{2}$

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Example Question #74 : Solving And Graphing Radicals

Solve: $\frac{1}{\sqrt{3x}} = 4$

Possible Answers:

$\frac{1}{24}$

$\frac{1}{72}$

$\frac{1}{48}$

$\frac{16}{3}$

Correct answer:

$\frac{1}{48}$

Explanation:

Multiply by $\sqrt{3x}$ on both sides.

$\frac{1}{\sqrt{3x}} \cdot \sqrt{3x}= 4 \cdot \sqrt{3x}$

$1=4\sqrt{3x}$

Divide by 4 on both sides.

$\frac{1}{4}=\frac{4\sqrt{3x}}{4}$

$\frac{1}{4} =\sqrt{3x}$

Square both sides to eliminate the radical.

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

$\frac{1}{16} = 3x$

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

$\frac{1}{16} \cdot \frac{1}{3}= 3x\cdot \frac{1}{3}$

The answer is: $\frac{1}{48}$

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Example Question #72 : Solving Radical Equations

Solve: $\frac{2}{\sqrt{8x}} = \sqrt{x}$

Possible Answers:

$\frac{\sqrt2}{4}$

$\frac{3\sqrt2}{4}$

$\frac{\sqrt2}{2}$

$\textup{No solution.}$

$\frac{\sqrt5}{8}$

Correct answer:

$\frac{\sqrt2}{2}$

Explanation:

Multiply both sides by the denominator on the left side.

$\frac{2}{\sqrt{8x}}\cdot \sqrt{8x} = \sqrt{x}\cdot \sqrt{8x}$

The equation becomes: $2 = \sqrt{8x^2}$

Square both sides to eliminate the radical.

$2 ^2= (\sqrt{8x^2})^2$

4=8x^2

Divide both sides by eight.

$\frac{4}{8}=\frac{8x^2}{8}$

$x= \pm\frac{1}{\sqrt{2}}$

Notice that the negative radical solution will not satisfy the original equation.

Rationalize the positive term.

$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt2}{2}$

The answer is: $\frac{\sqrt2}{2}$

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Example Question #76 : Solving And Graphing Radicals

Solve: $\sqrt[4]{32x+1} = 3$

Possible Answers:

$\frac{145}{32}$

$\frac{7}{2}$

$\frac{4}{3}$

$\frac{41}{16}$

$\frac{5}{2}$

Correct answer:

$\frac{5}{2}$

Explanation:

Raise both sides to the power of four.

$(\sqrt[4]{32x+1}) ^4= 3^4$

Simplify both sides.

32x+1 = 81

Subtract 1 from both sides.

32x+1 -1= 81-1

32x=80

Divide by 32 on both sides.

$\frac{32x}{32}=\frac{80}{32}$

Reduce this fraction.

$\frac{80}{32} = \frac{16\times 5}{16\times 2}$

The answer is: $\frac{5}{2}$

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Example Question #4291 : Algebra Ii

Solve the equation: $\sqrt[3]{8x-2} = 10$

Possible Answers:

128

125

$\frac{501}{4}$

$\frac{251}{4}$

Correct answer:

$\frac{501}{4}$

Explanation:

In order to eliminate the radical on both sides, we will need to cube both sides.

$(\sqrt[3]{8x-2})^3 = 10^3$

Simplify both sides.

8x-2 = 1000

Add two on both sides.

8x-2+2 = 1000+2

8x=1002

Divide by eight on both sides.

$\frac{8x}{8}=\frac{1002}{8}$

Reduce the fractions on both sides.

The answer is: $\frac{501}{4}$

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Example Question #392 : Radicals

Solve: $\sqrt{-5x+3} = 4$

Possible Answers:

$\textup{No solution.}$

$-\frac{13}{5}$

$-\frac{26}{5}$

Correct answer:

$-\frac{13}{5}$

Explanation:

Square both sides to eliminate the radical.

$(\sqrt{-5x+3} )^2= 4^2$

-5x+3 = 16

Subtract three on both sides.

-5x+3 -3= 16-3

The equation becomes:

-5x= 13

Divide by negative five on both sides.

$\frac{-5x}{-5}= \frac{13}{-5}$

The answer is: $-\frac{13}{5}$

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Example Question #79 : Solving And Graphing Radicals

Solve the equation: $\sqrt[4]{-3x}+8 = 11$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

Subtract eight from both sides.

$\sqrt[4]{-3x}+8-8 = 11-8$

$\sqrt[4]{-3x} =3$

Raise both sides by the power of four.

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

-3x =81

Divide both sides by three.

$\frac{-3x }{-3}=\frac{81}{-3}$

The answer is:

Report an Error

Example Question #80 : Solving And Graphing Radicals

Solve: $\sqrt{6x-9} -3 =5$

Possible Answers:

$\frac{73}{6}$

$\frac{53}{3}$

$\frac{13}{3}$

$\frac{13}{6}$

$\frac{53}{6}$

Correct answer:

$\frac{73}{6}$

Explanation:

Add three on both sides.

$\sqrt{6x-9} -3 +3=5+3$

$\sqrt{6x-9} = 8$

Square both sides.

$(\sqrt{6x-9})^2 = 8^2$

6x-9 = 64

Add nine on both sides.

6x-9+9 = 64+9

Simplify both sides.

6x=73

Divide by 6 on both sides.

$\frac{6x}{6}=\frac{73}{6}$

The answer is: $\frac{73}{6}$

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Algebra II : Solving Radical Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #71 : Solving Radical Equations

Example Question #72 : Solving And Graphing Radicals

Example Question #73 : Solving And Graphing Radicals

Example Question #74 : Solving And Graphing Radicals

Example Question #72 : Solving Radical Equations

Example Question #76 : Solving And Graphing Radicals

Example Question #4291 : Algebra Ii

Example Question #392 : Radicals

Example Question #79 : Solving And Graphing Radicals

Example Question #80 : Solving And Graphing Radicals

All Algebra II Resources

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