Solve for a ( fourth root of x)/(x^4)=x^a

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

Step 1

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

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

Step 2

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

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

Step 3

Move $x^{4}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 4

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 4$ by $4$ .

$x^{a} = x^{\frac{1 - 16}{4}}$

Subtract $16$ from $1$ .

$x^{a} = x^{\frac{- 15}{4}}$

Move the negative in front of the fraction.

$x^{a} = x^{- \frac{15}{4}}$

Step 5

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

$a = - \frac{15}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{15}{4}$

Decimal Form:

$a = - 3.75$

Mixed Number Form:

$a = - 3 \frac{3}{4}$