Solve for a fourth root of a-5=-1

$\sqrt[4]{a - 5} = - 1$

Step 1

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

${\sqrt[4]{a - 5}}^{4} = {(- 1)}^{4}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{a - 5}$ as ${(a - 5)}^{\frac{1}{4}}$ .

${({(a - 5)}^{\frac{1}{4}})}^{4} = {(- 1)}^{4}$

Simplify the left side.

Simplify ${({(a - 5)}^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${({(a - 5)}^{\frac{1}{4}})}^{4}$ .

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

${(a - 5)}^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(a - 5)}^{\frac{1}{4} \cdot 4} = {(- 1)}^{4}$

Rewrite the expression.

${(a - 5)}^{1} = {(- 1)}^{4}$

Simplify.

$a - 5 = {(- 1)}^{4}$

Simplify the right side.

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

$a - 5 = 1$

Step 3

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

Add $5$ to both sides of the equation.

$a = 1 + 5$

Add $1$ and $5$ .

$a = 6$

Step 4

Exclude the solutions that do not make $\sqrt[4]{a - 5} = - 1$ true.

$No solution$