Solve for a ((x^(1/4))^3)/( fourth root of x)=x^a

$\frac{{(x^{\frac{1}{4}})}^{3}}{\sqrt[4]{x}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{{(x^{\frac{1}{4}})}^{3}}{\sqrt[4]{x}}$ .

$x^{a} = \frac{{(x^{\frac{1}{4}})}^{3}}{\sqrt[4]{x}}$

Step 2

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

$x^{a} = \frac{{(x^{\frac{1}{4}})}^{3}}{x^{\frac{1}{4}}}$

Step 3

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

$x^{a} = \frac{x^{\frac{1}{4} \cdot 3}}{x^{\frac{1}{4}}}$

Step 4

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

$x^{a} = x^{\frac{1}{4} \cdot 3} x^{- \frac{1}{4}}$

Step 5

Multiply $x^{\frac{1}{4} \cdot 3}$ by $x^{- \frac{1}{4}}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{4} \cdot 3 - \frac{1}{4}}$

Combine $\frac{1}{4}$ and $3$ .

$x^{a} = x^{\frac{3}{4} - \frac{1}{4}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 - 1}{4}}$

Subtract $1$ from $3$ .

$x^{a} = x^{\frac{2}{4}}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x^{a} = x^{\frac{2 (1)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

$a = \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1}{2}$

Decimal Form:

$a = 0.5$