Solve by Factoring -2=3-7 fifth root of x^2

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $3$ from both sides of the equation.

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

Add $7 \sqrt[5]{x^{2}}$ to both sides of the equation.

$- 2 - 3 + 7 \sqrt[5]{x^{2}} = 0$

Step 2

Subtract $3$ from $- 2$ .

$- 5 + 7 \sqrt[5]{x^{2}} = 0$

Step 3

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

$- 5 + 7 x^{\frac{2}{5}} = 0$

Step 4

Add $5$ to both sides of the equation.

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

Step 5

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

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

Step 6

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 $7^{\frac{5}{2}} x$ .

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

Step 7

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.

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

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

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

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

Simplify the left side.

Cancel the common factor.

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

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.

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

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

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

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

Simplify the left side.

Cancel the common factor.

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

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.

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

Solve by Factoring -2=3-7 fifth root of x^2

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