Solve for a (x^(1/5))^2 fifth root of x=x^a

${(x^{\frac{1}{5}})}^{2} \sqrt[5]{x} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = {(x^{\frac{1}{5}})}^{2} \sqrt[5]{x}$ .

$x^{a} = {(x^{\frac{1}{5}})}^{2} \sqrt[5]{x}$

Step 2

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

$x^{a} = {(x^{\frac{1}{5}})}^{2} x^{\frac{1}{5}}$

Step 3

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

$x^{a} = x^{\frac{1}{5} \cdot 2} x^{\frac{1}{5}}$

Step 4

Multiply $x^{\frac{1}{5} \cdot 2}$ by $x^{\frac{1}{5}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{a} = x^{\frac{1}{5} \cdot 2 + \frac{1}{5}}$

Combine $\frac{1}{5}$ and $2$ .

$x^{a} = x^{\frac{2}{5} + \frac{1}{5}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{2 + 1}{5}}$

Add $2$ and $1$ .

$x^{a} = x^{\frac{3}{5}}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$a = \frac{3}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{5}$

Decimal Form:

$a = 0.6$