Solve the Rational Equation for n (n root of a)^n=a

${(\sqrt[n]{a})}^{n} = a$

Step 1

Rewrite ${\sqrt[n]{a}}^{n}$ as $a$ .

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

${(a^{\frac{1}{n}})}^{n} = a$

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

$a^{\frac{1}{n} n} = a$

Combine $\frac{1}{n}$ and $n$ .

$a^{\frac{n}{n}} = a$

Cancel the common factor of $n$ .

Cancel the common factor.

$a^{\frac{n}{n}} = a$

Rewrite the expression.

$a^{1} = a$

Simplify.

$a = a$

Step 2

Move all terms containing $a$ to the left side of the equation.

Subtract $a$ from both sides of the equation.

$a - a = 0$

Subtract $a$ from $a$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

The variable $n$ got canceled.

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$