Solve for a x^(1/5)x^4=x^a

$x^{\frac{1}{5}} x^{4} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = x^{\frac{1}{5}} x^{4}$ .

$x^{a} = x^{\frac{1}{5}} x^{4}$

Step 2

Multiply $x^{\frac{1}{5}}$ by $x^{4}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{5} + 4}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{a} = x^{\frac{1}{5} + 4 \cdot \frac{5}{5}}$

Combine $4$ and $\frac{5}{5}$ .

$x^{a} = x^{\frac{1}{5} + \frac{4 \cdot 5}{5}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{1 + 4 \cdot 5}{5}}$

Simplify the numerator.

Multiply $4$ by $5$ .

$x^{a} = x^{\frac{1 + 20}{5}}$

Add $1$ and $20$ .

$x^{a} = x^{\frac{21}{5}}$

Step 3

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

$a = \frac{21}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = \frac{21}{5}$

Decimal Form:

$a = 4.2$

Mixed Number Form:

$a = 4 \frac{1}{5}$