Solve for x 4 fifth root of (2x-4)^2=16

$4 \sqrt[5]{{(2 x - 4)}^{2}} = 16$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(4 \sqrt[5]{{(2 x - 4)}^{2}})}^{5} = 16^{5}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(2 x - 4)}^{2}}$ as ${(2 x - 4)}^{\frac{2}{5}}$ .

${(4 {(2 x - 4)}^{\frac{2}{5}})}^{5} = 16^{5}$

Simplify the left side.

Simplify ${(4 {(2 x - 4)}^{\frac{2}{5}})}^{5}$ .

Apply the product rule to $4 {(2 x - 4)}^{\frac{2}{5}}$ .

$4^{5} {({(2 x - 4)}^{\frac{2}{5}})}^{5} = 16^{5}$

Raise $4$ to the power of $5$ .

$1024 {({(2 x - 4)}^{\frac{2}{5}})}^{5} = 16^{5}$

Multiply the exponents in ${({(2 x - 4)}^{\frac{2}{5}})}^{5}$ .

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

$1024 {(2 x - 4)}^{\frac{2}{5} \cdot 5} = 16^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$1024 {(2 x - 4)}^{\frac{2}{5} \cdot 5} = 16^{5}$

Rewrite the expression.

$1024 {(2 x - 4)}^{2} = 16^{5}$

Simplify the right side.

Raise $16$ to the power of $5$ .

$1024 {(2 x - 4)}^{2} = 1048576$

Step 3

Solve for $x$ .

Divide each term in $1024 {(2 x - 4)}^{2} = 1048576$ by $1024$ and simplify.

Divide each term in $1024 {(2 x - 4)}^{2} = 1048576$ by $1024$ .

$\frac{1024 {(2 x - 4)}^{2}}{1024} = \frac{1048576}{1024}$

Simplify the left side.

Cancel the common factor of $1024$ .

Cancel the common factor.

$\frac{1024 {(2 x - 4)}^{2}}{1024} = \frac{1048576}{1024}$

Divide ${(2 x - 4)}^{2}$ by $1$ .

${(2 x - 4)}^{2} = \frac{1048576}{1024}$

Simplify the right side.

Divide $1048576$ by $1024$ .

${(2 x - 4)}^{2} = 1024$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$2 x - 4 = \pm \sqrt{1024}$

Simplify $\pm \sqrt{1024}$ .

Rewrite $1024$ as $32^{2}$ .

$2 x - 4 = \pm \sqrt{32^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$2 x - 4 = \pm 32$

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.

$2 x - 4 = 32$

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

Add $4$ to both sides of the equation.

$2 x = 32 + 4$

Add $32$ and $4$ .

$2 x = 36$

Divide each term in $2 x = 36$ by $2$ and simplify.

Divide each term in $2 x = 36$ by $2$ .

$\frac{2 x}{2} = \frac{36}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{36}{2}$

Divide $x$ by $1$ .

$x = \frac{36}{2}$

Simplify the right side.

Divide $36$ by $2$ .

$x = 18$

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

$2 x - 4 = - 32$

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

Add $4$ to both sides of the equation.

$2 x = - 32 + 4$

Add $- 32$ and $4$ .

$2 x = - 28$

Divide each term in $2 x = - 28$ by $2$ and simplify.

Divide each term in $2 x = - 28$ by $2$ .

$\frac{2 x}{2} = \frac{- 28}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 28}{2}$

Divide $x$ by $1$ .

$x = \frac{- 28}{2}$

Simplify the right side.

Divide $- 28$ by $2$ .

$x = - 14$

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

$x = 18, - 14$

Solve for x 4 fifth root of (2x-4)^2=16

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