Solve the Rational Equation for x -2=3-7 fifth root of x^2

$- 2 = 3 - 7 \sqrt[5]{x^{2}}$

Rewrite the equation as $3 - 7 \sqrt[5]{x^{2}} = - 2$ .

$3 - 7 \sqrt[5]{x^{2}} = - 2$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$3 - 7 x^{\frac{2}{5}} = - 2$

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

Subtract $3$ from both sides of the equation.

$- 7 x^{\frac{2}{5}} + 2 = - 3$

Subtract $2$ from both sides of the equation.

$- 7 x^{\frac{2}{5}} = - 3 - 2$

Subtract $2$ from $- 3$ .

$- 7 x^{\frac{2}{5}} = - 5$

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(- 7 x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm {(- 5)}^{\frac{5}{2}}$

Simplify the left side.

Simplify ${(- 7 x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

Apply the product rule to $- 7 x^{\frac{2}{5}}$ .

${(- 7)}^{\frac{5}{2}} {(x^{\frac{2}{5}})}^{\frac{5}{2}} = \pm {(- 5)}^{\frac{5}{2}}$

Multiply the exponents in ${(x^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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

${(- 7)}^{\frac{5}{2}} x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(- 5)}^{\frac{5}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 7)}^{\frac{5}{2}} x^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(- 5)}^{\frac{5}{2}}$

Rewrite the expression.

${(- 7)}^{\frac{5}{2}} x^{\frac{1}{5} \cdot 5} = \pm {(- 5)}^{\frac{5}{2}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(- 7)}^{\frac{5}{2}} x^{\frac{1}{5} \cdot 5} = \pm {(- 5)}^{\frac{5}{2}}$

Rewrite the expression.

${(- 7)}^{\frac{5}{2}} x^{1} = \pm {(- 5)}^{\frac{5}{2}}$

Simplify.

${(- 7)}^{\frac{5}{2}} x = \pm {(- 5)}^{\frac{5}{2}}$

Reorder factors in ${(- 7)}^{\frac{5}{2}} x$ .

$x {(- 7)}^{\frac{5}{2}} = \pm {(- 5)}^{\frac{5}{2}}$

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.

$x {(- 7)}^{\frac{5}{2}} = {(- 5)}^{\frac{5}{2}}$

Divide each term in $x {(- 7)}^{\frac{5}{2}} = {(- 5)}^{\frac{5}{2}}$ by ${(- 7)}^{\frac{5}{2}}$ and simplify.

Divide each term in $x {(- 7)}^{\frac{5}{2}} = {(- 5)}^{\frac{5}{2}}$ by ${(- 7)}^{\frac{5}{2}}$ .

$\frac{x {(- 7)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}} = \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x {(- 7)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}} = \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Divide $x$ by $1$ .

$x = \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

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

$x {(- 7)}^{\frac{5}{2}} = - {(- 5)}^{\frac{5}{2}}$

Divide each term in $x {(- 7)}^{\frac{5}{2}} = - {(- 5)}^{\frac{5}{2}}$ by ${(- 7)}^{\frac{5}{2}}$ and simplify.

Divide each term in $x {(- 7)}^{\frac{5}{2}} = - {(- 5)}^{\frac{5}{2}}$ by ${(- 7)}^{\frac{5}{2}}$ .

$\frac{x {(- 7)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}} = \frac{- {(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x {(- 7)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}} = \frac{- {(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Divide $x$ by $1$ .

$x = \frac{- {(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

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

$x = \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}, - \frac{{(- 5)}^{\frac{5}{2}}}{{(- 7)}^{\frac{5}{2}}}$

Exclude the solutions that do not make $- 2 = 3 - 7 \sqrt[5]{x^{2}}$ true.

$No solution$

Solve the Rational Equation for x -2=3-7 fifth root of x^2

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