Solve for x 6x^(2/3)-x^(1/3)=12

$6 x^{\frac{2}{3}} - x^{\frac{1}{3}} = 12$

Step 1

Find a common factor $x^{\frac{1}{3}}$ that is present in each term.

$6 {(x^{\frac{1}{3}})}^{2} - x^{\frac{1}{3}} - 12$

Step 2

Substitute $u$ for $x^{\frac{1}{3}}$ .

$6 {(u)}^{2} - (u) - 12 = 0$

Step 3

Solve for $u$ .

Multiply $- 1$ by $u$ .

$6 u^{2} - u - 12 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 12 = - 72$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- u$ .

$6 u^{2} - (u) - 12 = 0$

Rewrite $- 1$ as $8$ plus $- 9$

$6 u^{2} + (8 - 9) u - 12 = 0$

Apply the distributive property.

$6 u^{2} + 8 u - 9 u - 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 u^{2} + 8 u) - 9 u - 12 = 0$

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

$2 u (3 u + 4) - 3 (3 u + 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 u + 4$ .

$(3 u + 4) (2 u - 3) = 0$

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

$3 u + 4 = 0$

$2 u - 3 = 0$

Set $3 u + 4$ equal to $0$ and solve for $u$ .

Set $3 u + 4$ equal to $0$ .

$3 u + 4 = 0$

Solve $3 u + 4 = 0$ for $u$ .

Subtract $4$ from both sides of the equation.

$3 u = - 4$

Divide each term in $3 u = - 4$ by $3$ and simplify.

Divide each term in $3 u = - 4$ by $3$ .

$\frac{3 u}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 u}{3} = \frac{- 4}{3}$

Divide $u$ by $1$ .

$u = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$u = - \frac{4}{3}$

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

Set $2 u - 3$ equal to $0$ .

$2 u - 3 = 0$

Solve $2 u - 3 = 0$ for $u$ .

Add $3$ to both sides of the equation.

$2 u = 3$

Divide each term in $2 u = 3$ by $2$ and simplify.

Divide each term in $2 u = 3$ by $2$ .

$\frac{2 u}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 u}{2} = \frac{3}{2}$

Divide $u$ by $1$ .

$u = \frac{3}{2}$

The final solution is all the values that make $(3 u + 4) (2 u - 3) = 0$ true.

$u = - \frac{4}{3}, \frac{3}{2}$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{3}} = - \frac{4}{3}, \frac{3}{2}$

Step 5

Solve for $x^{\frac{1}{3}} = - \frac{4}{3}$ for $x$ .

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

Apply the product rule to $- \frac{4}{3}$ .

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

Apply the product rule to $\frac{4}{3}$ .

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

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

$x = - \frac{4^{3}}{3^{3}}$

Raise $4$ to the power of $3$ .

$x = - \frac{64}{3^{3}}$

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

$x = - \frac{64}{27}$

Step 6

Solve for $x^{\frac{1}{3}} = \frac{3}{2}$ for $x$ .

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

$x = \frac{27}{2^{3}}$

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

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

Step 7

List all of the solutions.

$x = - \frac{64}{27}, \frac{27}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{64}{27}, \frac{27}{8}$

Decimal Form:

$x = - 2.\overline{370}, 3.375$

Mixed Number Form:

$x = - 2 \frac{10}{27}, 3 \frac{3}{8}$

Solve for x 6x^(2/3)-x^(1/3)=12

Related Questions

Support LCMGCF.com by disabling your adblocker.