Solve for x 6 cube root of x-3+2=1/2

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

Step 1

Move all terms not containing $6 \sqrt[3]{x - 3}$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 2$ by $2$ .

$6 \sqrt[3]{x - 3} = \frac{1 - 4}{2}$

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

Step 2

To remove the radical on the left side of the equation, cube both sides of the equation.

${(6 \sqrt[3]{x - 3})}^{3} = {(- \frac{3}{2})}^{3}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${(6 {(x - 3)}^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $6 {(x - 3)}^{\frac{1}{3}}$ .

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

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

$216 {({(x - 3)}^{\frac{1}{3}})}^{3} = {(- \frac{3}{2})}^{3}$

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

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

$216 {(x - 3)}^{\frac{1}{3} \cdot 3} = {(- \frac{3}{2})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$216 {(x - 3)}^{\frac{1}{3} \cdot 3} = {(- \frac{3}{2})}^{3}$

Rewrite the expression.

$216 {(x - 3)}^{1} = {(- \frac{3}{2})}^{3}$

Simplify.

$216 (x - 3) = {(- \frac{3}{2})}^{3}$

Apply the distributive property.

$216 x + 216 \cdot - 3 = {(- \frac{3}{2})}^{3}$

Multiply $216$ by $- 3$ .

$216 x - 648 = {(- \frac{3}{2})}^{3}$

Simplify the right side.

Simplify ${(- \frac{3}{2})}^{3}$ .

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3}{2}$ .

$216 x - 648 = {(- 1)}^{3} {(\frac{3}{2})}^{3}$

Apply the product rule to $\frac{3}{2}$ .

$216 x - 648 = {(- 1)}^{3} \frac{3^{3}}{2^{3}}$

Raise $- 1$ to the power of $3$ .

$216 x - 648 = - \frac{3^{3}}{2^{3}}$

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

$216 x - 648 = - \frac{27}{2^{3}}$

Raise $2$ to the power of $3$ .

$216 x - 648 = - \frac{27}{8}$

Step 4

Solve for $x$ .

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

Add $648$ to both sides of the equation.

$216 x = - \frac{27}{8} + 648$

To write $648$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$216 x = - \frac{27}{8} + 648 \cdot \frac{8}{8}$

Combine $648$ and $\frac{8}{8}$ .

$216 x = - \frac{27}{8} + \frac{648 \cdot 8}{8}$

Combine the numerators over the common denominator.

$216 x = \frac{- 27 + 648 \cdot 8}{8}$

Simplify the numerator.

Multiply $648$ by $8$ .

$216 x = \frac{- 27 + 5184}{8}$

Add $- 27$ and $5184$ .

$216 x = \frac{5157}{8}$

Divide each term in $216 x = \frac{5157}{8}$ by $216$ and simplify.

Divide each term in $216 x = \frac{5157}{8}$ by $216$ .

$\frac{216 x}{216} = \frac{\frac{5157}{8}}{216}$

Simplify the left side.

Cancel the common factor of $216$ .

Cancel the common factor.

$\frac{216 x}{216} = \frac{\frac{5157}{8}}{216}$

Divide $x$ by $1$ .

$x = \frac{\frac{5157}{8}}{216}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5157}{8} \cdot \frac{1}{216}$

Cancel the common factor of $27$ .

Factor $27$ out of $5157$ .

$x = \frac{27 (191)}{8} \cdot \frac{1}{216}$

Factor $27$ out of $216$ .

$x = \frac{27 \cdot 191}{8} \cdot \frac{1}{27 \cdot 8}$

Cancel the common factor.

$x = \frac{27 \cdot 191}{8} \cdot \frac{1}{27 \cdot 8}$

Rewrite the expression.

$x = \frac{191}{8} \cdot \frac{1}{8}$

Multiply $\frac{191}{8}$ by $\frac{1}{8}$ .

$x = \frac{191}{8 \cdot 8}$

Multiply $8$ by $8$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.984375$

Mixed Number Form:

$x = 2 \frac{63}{64}$

Solve for x 6 cube root of x-3+2=1/2

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