Algebra Examples

$6 \sqrt[3]{x - 3} + 2 = \frac{1}{2}$

Step 1

Subtract $\frac{1}{2}$ from both sides of the equation.

$6 \sqrt[3]{x - 3} + 2 - \frac{1}{2} = 0$

Step 2

Simplify $6 \sqrt[3]{x - 3} + 2 - \frac{1}{2}$ .

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Step 2.1

To write $6 \sqrt[3]{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 \sqrt[3]{x - 3} \cdot \frac{2}{2} - \frac{1}{2} + 2 = 0$

Step 2.2

Combine $6 \sqrt[3]{x - 3}$ and $\frac{2}{2}$ .

$\frac{6 \sqrt[3]{x - 3} \cdot 2}{2} - \frac{1}{2} + 2 = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{6 \sqrt[3]{x - 3} \cdot 2 - 1}{2} + 2 = 0$

Step 2.4

Multiply $2$ by $6$ .

$\frac{12 \sqrt[3]{x - 3} - 1}{2} + 2 = 0$

Step 3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{12 \sqrt[3]{x - 3} - 1}{2} + 2 \cdot \frac{2}{2} = 0$

Step 4

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

$\frac{12 \sqrt[3]{x - 3} - 1}{2} + \frac{2 \cdot 2}{2} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{12 \sqrt[3]{x - 3} - 1 + 2 \cdot 2}{2} = 0$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{12 {(x - 3)}^{\frac{1}{3}} - 1 + 2 \cdot 2}{2} = 0$

Step 6.2

Multiply $2$ by $2$ .

$\frac{12 {(x - 3)}^{\frac{1}{3}} - 1 + 4}{2} = 0$

Step 6.3

Add $- 1$ and $4$ .

$\frac{12 {(x - 3)}^{\frac{1}{3}} + 3}{2} = 0$

Step 6.4

Factor $3$ out of $12 {(x - 3)}^{\frac{1}{3}} + 3$ .

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Step 6.4.1

Factor $3$ out of $12 {(x - 3)}^{\frac{1}{3}}$ .

$\frac{3 (4 {(x - 3)}^{\frac{1}{3}}) + 3}{2} = 0$

Step 6.4.2

Factor $3$ out of $3$ .

$\frac{3 (4 {(x - 3)}^{\frac{1}{3}}) + 3 (1)}{2} = 0$

Step 6.4.3

Factor $3$ out of $3 (4 {(x - 3)}^{\frac{1}{3}}) + 3 (1)$ .

$\frac{3 (4 {(x - 3)}^{\frac{1}{3}} + 1)}{2} = 0$

Step 7

Set the numerator equal to zero.

$3 (4 {(x - 3)}^{\frac{1}{3}} + 1) = 0$

Step 8

Solve the equation for $x$ .

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Step 8.1

Simplify $3 (4 {(x - 3)}^{\frac{1}{3}} + 1)$ .

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Step 8.1.1

Apply the distributive property.

$3 (4 {(x - 3)}^{\frac{1}{3}}) + 3 \cdot 1 = 0$

Step 8.1.2

Multiply.

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Step 8.1.2.1

Multiply $4$ by $3$ .

$12 {(x - 3)}^{\frac{1}{3}} + 3 \cdot 1 = 0$

Step 8.1.2.2

Multiply $3$ by $1$ .

$12 {(x - 3)}^{\frac{1}{3}} + 3 = 0$

Step 8.2

Subtract $3$ from both sides of the equation.

$12 {(x - 3)}^{\frac{1}{3}} = - 3$

Step 8.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(12 {(x - 3)}^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Step 8.4

Simplify the exponent.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Simplify ${(12 {(x - 3)}^{\frac{1}{3}})}^{3}$ .

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Step 8.4.1.1.1

Apply the product rule to $12 {(x - 3)}^{\frac{1}{3}}$ .

$12^{3} {({(x - 3)}^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Step 8.4.1.1.2

Raise $12$ to the power of $3$ .

$1728 {({(x - 3)}^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Step 8.4.1.1.3

Multiply the exponents in ${({(x - 3)}^{\frac{1}{3}})}^{3}$ .

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Step 8.4.1.1.3.1

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

$1728 {(x - 3)}^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 8.4.1.1.3.2

Cancel the common factor of $3$ .

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Step 8.4.1.1.3.2.1

Cancel the common factor.

$1728 {(x - 3)}^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Step 8.4.1.1.3.2.2

Rewrite the expression.

$1728 {(x - 3)}^{1} = {(- 3)}^{3}$

Step 8.4.1.1.4

Simplify.

$1728 (x - 3) = {(- 3)}^{3}$

Step 8.4.1.1.5

Apply the distributive property.

$1728 x + 1728 \cdot - 3 = {(- 3)}^{3}$

Step 8.4.1.1.6

Multiply $1728$ by $- 3$ .

$1728 x - 5184 = {(- 3)}^{3}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

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

$1728 x - 5184 = - 27$

Step 8.5

Solve for $x$ .

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Step 8.5.1

Move all terms not containing $x$ to the right side of the equation.

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Step 8.5.1.1

Add $5184$ to both sides of the equation.

$1728 x = - 27 + 5184$

Step 8.5.1.2

Add $- 27$ and $5184$ .

$1728 x = 5157$

Step 8.5.2

Divide each term in $1728 x = 5157$ by $1728$ and simplify.

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Step 8.5.2.1

Divide each term in $1728 x = 5157$ by $1728$ .

$\frac{1728 x}{1728} = \frac{5157}{1728}$

Step 8.5.2.2

Simplify the left side.

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Step 8.5.2.2.1

Cancel the common factor of $1728$ .

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Step 8.5.2.2.1.1

Cancel the common factor.

$\frac{1728 x}{1728} = \frac{5157}{1728}$

Step 8.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5157}{1728}$

Step 8.5.2.3

Simplify the right side.

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Step 8.5.2.3.1

Cancel the common factor of $5157$ and $1728$ .

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Step 8.5.2.3.1.1

Factor $27$ out of $5157$ .

$x = \frac{27 (191)}{1728}$

Step 8.5.2.3.1.2

Cancel the common factors.

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Step 8.5.2.3.1.2.1

Factor $27$ out of $1728$ .

$x = \frac{27 \cdot 191}{27 \cdot 64}$

Step 8.5.2.3.1.2.2

Cancel the common factor.

$x = \frac{27 \cdot 191}{27 \cdot 64}$

Step 8.5.2.3.1.2.3

Rewrite the expression.

$x = \frac{191}{64}$