Solve for x 12x^5-384x^(10/3)=0

$12 x^{5} - 384 x^{\frac{10}{3}} = 0$

Find a common factor $x^{\frac{10}{3}}$ that is present in each term.

$12 {(x^{\frac{10}{3}})}^{\frac{3}{2}} - 384 x^{\frac{10}{3}}$

Substitute $u$ for $x^{\frac{10}{3}}$ .

$12 {(u)}^{\frac{3}{2}} - 384 u = 0$

Solve for $u$ .

Factor $12 u$ out of $12 u^{\frac{3}{2}} - 384 u$ .

Factor $12 u$ out of $12 u^{\frac{3}{2}}$ .

$12 u (u^{\frac{1}{2}}) - 384 u = 0$

Factor $12 u$ out of $- 384 u$ .

$12 u (u^{\frac{1}{2}}) + 12 u (- 32) = 0$

Factor $12 u$ out of $12 u (u^{\frac{1}{2}}) + 12 u (- 32)$ .

$12 u (u^{\frac{1}{2}} - 32) = 0$

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

$u = 0$

$u^{\frac{1}{2}} - 32 = 0$

Set $u$ equal to $0$ .

$u = 0$

Set $u^{\frac{1}{2}} - 32$ equal to $0$ and solve for $u$ .

Set $u^{\frac{1}{2}} - 32$ equal to $0$ .

$u^{\frac{1}{2}} - 32 = 0$

Solve $u^{\frac{1}{2}} - 32 = 0$ for $u$ .

Add $32$ to both sides of the equation.

$u^{\frac{1}{2}} = 32$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{2}})}^{2} = 32^{2}$

Simplify the exponent.

Simplify the left side.

Simplify ${(u^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(u^{\frac{1}{2}})}^{2}$ .

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

$u^{\frac{1}{2} \cdot 2} = 32^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$u^{\frac{1}{2} \cdot 2} = 32^{2}$

Rewrite the expression.

$u^{1} = 32^{2}$

Simplify.

$u = 32^{2}$

Simplify the right side.

Raise $32$ to the power of $2$ .

$u = 1024$

The final solution is all the values that make $12 u (u^{\frac{1}{2}} - 32) = 0$ true.

$u = 0, 1024$

Substitute $x$ for $u$ .

$x^{\frac{10}{3}} = 0, 1024$

Solve for $x^{\frac{10}{3}} = 0$ for $x$ .

Raise each side of the equation to the power of $\frac{3}{10}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{10}{3}})}^{\frac{3}{10}} = \pm 0^{\frac{3}{10}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{10}{3}})}^{\frac{3}{10}}$ .

Multiply the exponents in ${(x^{\frac{10}{3}})}^{\frac{3}{10}}$ .

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

$x^{\frac{10}{3} \cdot \frac{3}{10}} = \pm 0^{\frac{3}{10}}$

Cancel the common factor of $10$ .

Cancel the common factor.

$x^{\frac{10}{3} \cdot \frac{3}{10}} = \pm 0^{\frac{3}{10}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{10}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{10}}$

Rewrite the expression.

$x^{1} = \pm 0^{\frac{3}{10}}$

Simplify.

$x = \pm 0^{\frac{3}{10}}$

Simplify the right side.

Simplify $\pm 0^{\frac{3}{10}}$ .

Simplify the expression.

Rewrite $0$ as $0^{10}$ .

$x = \pm {(0^{10})}^{\frac{3}{10}}$

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

$x = \pm 0^{10 (\frac{3}{10})}$

Cancel the common factor of $10$ .

Cancel the common factor.

$x = \pm 0^{10 (\frac{3}{10})}$

Rewrite the expression.

$x = \pm 0^{3}$

Raising $0$ to any positive power yields $0$ .

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Solve for $x^{\frac{10}{3}} = 1024$ for $x$ .

Raise each side of the equation to the power of $\frac{3}{10}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{10}{3}})}^{\frac{3}{10}} = \pm 1024^{\frac{3}{10}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{10}{3}})}^{\frac{3}{10}}$ .

Multiply the exponents in ${(x^{\frac{10}{3}})}^{\frac{3}{10}}$ .

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

$x^{\frac{10}{3} \cdot \frac{3}{10}} = \pm 1024^{\frac{3}{10}}$

Cancel the common factor of $10$ .

Cancel the common factor.

$x^{\frac{10}{3} \cdot \frac{3}{10}} = \pm 1024^{\frac{3}{10}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 1024^{\frac{3}{10}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 1024^{\frac{3}{10}}$

Rewrite the expression.

$x^{1} = \pm 1024^{\frac{3}{10}}$

Simplify.

$x = \pm 1024^{\frac{3}{10}}$

Simplify the right side.

Simplify $\pm 1024^{\frac{3}{10}}$ .

Simplify the expression.

Rewrite $1024$ as $2^{10}$ .

$x = \pm {(2^{10})}^{\frac{3}{10}}$

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

$x = \pm 2^{10 (\frac{3}{10})}$

Cancel the common factor of $10$ .

Cancel the common factor.

$x = \pm 2^{10 (\frac{3}{10})}$

Rewrite the expression.

$x = \pm 2^{3}$

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

$x = \pm 8$

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

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

$x = 8$

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

$x = - 8$

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

$x = 8, - 8$

List all of the solutions.

$x = 0, 8, - 8$

Exclude the solutions that do not make $12 x^{5} - 384 x^{\frac{10}{3}} = 0$ true.

$x = 0, 8$

Solve for x 12x^5-384x^(10/3)=0

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