Solve for a x^5 fourth root of x=x^a

$x^{5} \sqrt[4]{x} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = x^{5} \sqrt[4]{x}$ .

$x^{a} = x^{5} \sqrt[4]{x}$

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $5$ by $4$ .

$x^{a} = x^{\frac{20 + 1}{4}}$

Add $20$ and $1$ .

$x^{a} = x^{\frac{21}{4}}$

Step 4

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

$a = \frac{21}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{21}{4}$

Decimal Form:

$a = 5.25$

Mixed Number Form:

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