Algebra Examples

$x^{\frac{1}{2}} - 3 x^{\frac{1}{3}} = 3 x^{\frac{1}{6}} - 9$

Step 1

Find a common factor $x^{\frac{1}{6}}$ that is present in each term.

${(x^{\frac{1}{6}})}^{3} - 3 {(x^{\frac{1}{6}})}^{2} - 3 x^{\frac{1}{6}} + 9$

Step 2

Substitute $u$ for $x^{\frac{1}{6}}$ .

${(u)}^{3} - 3 {(u)}^{2} - 3 u + 9 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Remove parentheses.

$u^{3} - 3 u^{2} - 3 u + 9 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor out the greatest common factor from each group.

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Step 3.2.1.1

Group the first two terms and the last two terms.

$(u^{3} - 3 u^{2}) - 3 u + 9 = 0$

Step 3.2.1.2

Factor out the greatest common factor (GCF) from each group.

$u^{2} (u - 3) - 3 (u - 3) = 0$

Step 3.2.2

Factor the polynomial by factoring out the greatest common factor, $u - 3$ .

$(u - 3) (u^{2} - 3) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 3 = 0$

$u^{2} - 3 = 0$

Step 3.4

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 3.4.2

Add $3$ to both sides of the equation.

$u = 3$

Step 3.5

Set $u^{2} - 3$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u^{2} - 3$ equal to $0$ .

$u^{2} - 3 = 0$

Step 3.5.2

Solve $u^{2} - 3 = 0$ for $u$ .

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Step 3.5.2.1

Add $3$ to both sides of the equation.

$u^{2} = 3$

Step 3.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{3}$

Step 3.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$u = \sqrt{3}$

Step 3.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$u = - \sqrt{3}$

Step 3.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$u = \sqrt{3}, - \sqrt{3}$

Step 3.6

The final solution is all the values that make $(u - 3) (u^{2} - 3) = 0$ true.

$u = 3, \sqrt{3}, - \sqrt{3}$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{6}} = 3, \sqrt{3}, - \sqrt{3}$

Step 5

Solve for $x^{\frac{1}{6}} = 3$ for $x$ .

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Step 5.1

Raise each side of the equation to the power of $6$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{6}})}^{6} = 3^{6}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify ${(x^{\frac{1}{6}})}^{6}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{6}$ .

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Step 5.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{6} \cdot 6} = 3^{6}$

Step 5.2.1.1.1.2

Cancel the common factor of $6$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{6} \cdot 6} = 3^{6}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = 3^{6}$

Step 5.2.1.1.2

Simplify.

$x = 3^{6}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Raise $3$ to the power of $6$ .

$x = 729$

Step 6

Solve for $x^{\frac{1}{6}} = \sqrt{3}$ for $x$ .

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Step 6.1

Raise each side of the equation to the power of $6$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{6}})}^{6} = {\sqrt{3}}^{6}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify ${(x^{\frac{1}{6}})}^{6}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{6}$ .

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Step 6.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{6} \cdot 6} = {\sqrt{3}}^{6}$

Step 6.2.1.1.1.2

Cancel the common factor of $6$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{6} \cdot 6} = {\sqrt{3}}^{6}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$x^{1} = {\sqrt{3}}^{6}$

Step 6.2.1.1.2

Simplify.

$x = {\sqrt{3}}^{6}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify ${\sqrt{3}}^{6}$ .

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Step 6.2.2.1.1

Rewrite ${\sqrt{3}}^{6}$ as $3^{3}$ .

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Step 6.2.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = {(3^{\frac{1}{2}})}^{6}$

Step 6.2.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 3^{\frac{1}{2} \cdot 6}$

Step 6.2.2.1.1.3

Combine $\frac{1}{2}$ and $6$ .

$x = 3^{\frac{6}{2}}$

Step 6.2.2.1.1.4

Cancel the common factor of $6$ and $2$ .

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Step 6.2.2.1.1.4.1

Factor $2$ out of $6$ .

$x = 3^{\frac{2 \cdot 3}{2}}$

Step 6.2.2.1.1.4.2

Cancel the common factors.

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Step 6.2.2.1.1.4.2.1

Factor $2$ out of $2$ .

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

Step 6.2.2.1.1.4.2.2

Cancel the common factor.

$x = 3^{\frac{2 \cdot 3}{2 \cdot 1}}$

Step 6.2.2.1.1.4.2.3

Rewrite the expression.

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

Step 6.2.2.1.1.4.2.4

Divide $3$ by $1$ .

$x = 3^{3}$

Step 6.2.2.1.2

Raise $3$ to the power of $3$ .

$x = 27$

Step 7

List all of the solutions.

$x = 729, 27, 27$