Solve the Rational Equation for x 8- fourth root of x=7

$8 - \sqrt[4]{x} = 7$

Step 1

Move all terms not containing $- \sqrt[4]{x}$ to the right side of the equation.

Subtract $8$ from both sides of the equation.

$- \sqrt[4]{x} = 7 - 8$

Subtract $8$ from $7$ .

$- \sqrt[4]{x} = - 1$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${(- \sqrt[4]{x})}^{4} = {(- 1)}^{4}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

${(- x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Simplify the left side.

Simplify ${(- x^{\frac{1}{4}})}^{4}$ .

Apply the product rule to $- x^{\frac{1}{4}}$ .

${(- 1)}^{4} {(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Raise $- 1$ to the power of $4$ .

$1 {(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Multiply ${(x^{\frac{1}{4}})}^{4}$ by $1$ .

${(x^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Rewrite the expression.

$x^{1} = {(- 1)}^{4}$

Simplify.

$x = {(- 1)}^{4}$

Simplify the right side.

Raise $- 1$ to the power of $4$ .

$x = 1$