Solve for x 7x^(12/7)-448x^(6/7)=0

$7 x^{\frac{12}{7}} - 448 x^{\frac{6}{7}} = 0$

Step 1

Factor the left side of the equation.

Rewrite $x^{\frac{12}{7}}$ as ${(x^{\frac{6}{7}})}^{2}$ .

$7 {(x^{\frac{6}{7}})}^{2} - 448 x^{\frac{6}{7}} = 0$

Let $u = x^{\frac{6}{7}}$ . Substitute $u$ for all occurrences of $x^{\frac{6}{7}}$ .

$7 u^{2} - 448 u = 0$

Factor $7 u$ out of $7 u^{2} - 448 u$ .

Factor $7 u$ out of $7 u^{2}$ .

$7 u (u) - 448 u = 0$

Factor $7 u$ out of $- 448 u$ .

$7 u (u) + 7 u (- 64) = 0$

Factor $7 u$ out of $7 u (u) + 7 u (- 64)$ .

$7 u (u - 64) = 0$

Replace all occurrences of $u$ with $x^{\frac{6}{7}}$ .

$7 x^{\frac{6}{7}} (x^{\frac{6}{7}} - 64) = 0$

Step 2

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

$x^{\frac{6}{7}} = 0$

$x^{\frac{6}{7}} - 64 = 0$

Step 3

Set $x^{\frac{6}{7}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{6}{7}}$ equal to $0$ .

$x^{\frac{6}{7}} = 0$

Solve $x^{\frac{6}{7}} = 0$ for $x$ .

Raise each side of the equation to the power of $\frac{7}{6}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{6}{7}})}^{\frac{7}{6}} = \pm 0^{\frac{7}{6}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{6}{7}})}^{\frac{7}{6}}$ .

Multiply the exponents in ${(x^{\frac{6}{7}})}^{\frac{7}{6}}$ .

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

$x^{\frac{6}{7} \cdot \frac{7}{6}} = \pm 0^{\frac{7}{6}}$

Cancel the common factor of $6$ .

Cancel the common factor.

$x^{\frac{6}{7} \cdot \frac{7}{6}} = \pm 0^{\frac{7}{6}}$

Rewrite the expression.

$x^{\frac{1}{7} \cdot 7} = \pm 0^{\frac{7}{6}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = \pm 0^{\frac{7}{6}}$

Rewrite the expression.

$x^{1} = \pm 0^{\frac{7}{6}}$

Simplify.

$x = \pm 0^{\frac{7}{6}}$

Simplify the right side.

Simplify $\pm 0^{\frac{7}{6}}$ .

Simplify the expression.

Rewrite $0$ as $0^{6}$ .

$x = \pm {(0^{6})}^{\frac{7}{6}}$

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

$x = \pm 0^{6 (\frac{7}{6})}$

Cancel the common factor of $6$ .

Cancel the common factor.

$x = \pm 0^{6 (\frac{7}{6})}$

Rewrite the expression.

$x = \pm 0^{7}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

Set $x^{\frac{6}{7}} - 64$ equal to $0$ and solve for $x$ .

Set $x^{\frac{6}{7}} - 64$ equal to $0$ .

$x^{\frac{6}{7}} - 64 = 0$

Solve $x^{\frac{6}{7}} - 64 = 0$ for $x$ .

Add $64$ to both sides of the equation.

$x^{\frac{6}{7}} = 64$

Raise each side of the equation to the power of $\frac{7}{6}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{6}{7}})}^{\frac{7}{6}} = \pm 64^{\frac{7}{6}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{6}{7}})}^{\frac{7}{6}}$ .

Multiply the exponents in ${(x^{\frac{6}{7}})}^{\frac{7}{6}}$ .

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

$x^{\frac{6}{7} \cdot \frac{7}{6}} = \pm 64^{\frac{7}{6}}$

Cancel the common factor of $6$ .

Cancel the common factor.

$x^{\frac{6}{7} \cdot \frac{7}{6}} = \pm 64^{\frac{7}{6}}$

Rewrite the expression.

$x^{\frac{1}{7} \cdot 7} = \pm 64^{\frac{7}{6}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = \pm 64^{\frac{7}{6}}$

Rewrite the expression.

$x^{1} = \pm 64^{\frac{7}{6}}$

Simplify.

$x = \pm 64^{\frac{7}{6}}$

Simplify the right side.

Simplify $\pm 64^{\frac{7}{6}}$ .

Simplify the expression.

Rewrite $64$ as $2^{6}$ .

$x = \pm {(2^{6})}^{\frac{7}{6}}$

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

$x = \pm 2^{6 (\frac{7}{6})}$

Cancel the common factor of $6$ .

Cancel the common factor.

$x = \pm 2^{6 (\frac{7}{6})}$

Rewrite the expression.

$x = \pm 2^{7}$

Raise $2$ to the power of $7$ .

$x = \pm 128$

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.

$x = 128$

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

$x = - 128$

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

$x = 128, - 128$

Step 5

The final solution is all the values that make $7 x^{\frac{6}{7}} (x^{\frac{6}{7}} - 64) = 0$ true.

$x = 0, 128, - 128$

Solve for x 7x^(12/7)-448x^(6/7)=0

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