Solve for a sixth root of 1-a^2=0

$\sqrt[6]{1 - a^{2}} = 0$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $6$ .

${\sqrt[6]{1 - a^{2}}}^{6} = 0^{6}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{1 - a^{2}}$ as ${(1 - a^{2})}^{\frac{1}{6}}$ .

${({(1 - a^{2})}^{\frac{1}{6}})}^{6} = 0^{6}$

Simplify the left side.

Simplify ${({(1 - a^{2})}^{\frac{1}{6}})}^{6}$ .

Multiply the exponents in ${({(1 - a^{2})}^{\frac{1}{6}})}^{6}$ .

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

${(1 - a^{2})}^{\frac{1}{6} \cdot 6} = 0^{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

${(1 - a^{2})}^{\frac{1}{6} \cdot 6} = 0^{6}$

Rewrite the expression.

${(1 - a^{2})}^{1} = 0^{6}$

Simplify.

$1 - a^{2} = 0^{6}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$1 - a^{2} = 0$

Step 3

Solve for $a$ .

Subtract $1$ from both sides of the equation.

$- a^{2} = - 1$

Divide each term in $- a^{2} = - 1$ by $- 1$ and simplify.

Divide each term in $- a^{2} = - 1$ by $- 1$ .

$\frac{- a^{2}}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{a^{2}}{1} = \frac{- 1}{- 1}$

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$a^{2} = 1$

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

$a = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$a = \pm 1$

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.

$a = 1$

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

$a = - 1$

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

$a = 1, - 1$

Solve for a sixth root of 1-a^2=0

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