Solve for a ( square root of x^3)/(x^(1/6))=x^a

$\frac{\sqrt{x^{3}}}{x^{\frac{1}{6}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt{x^{3}}}{x^{\frac{1}{6}}}$ .

$x^{a} = \frac{\sqrt{x^{3}}}{x^{\frac{1}{6}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$x^{a} = \frac{x^{\frac{3}{2}}}{x^{\frac{1}{6}}}$

Step 3

Move $x^{\frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{3}{2}} x^{- \frac{1}{6}}$

Step 4

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{1}{6}}$ by adding the exponents.

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

$x^{a} = x^{\frac{3}{2} - \frac{1}{6}}$

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{\frac{3}{2} \cdot \frac{3}{3} - \frac{1}{6}}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{3 \cdot 3}{2 \cdot 3} - \frac{1}{6}}$

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{3 \cdot 3}{6} - \frac{1}{6}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 \cdot 3 - 1}{6}}$

Simplify the numerator.

Multiply $3$ by $3$ .

$x^{a} = x^{\frac{9 - 1}{6}}$

Subtract $1$ from $9$ .

$x^{a} = x^{\frac{8}{6}}$

Cancel the common factor of $8$ and $6$ .

Factor $2$ out of $8$ .

$x^{a} = x^{\frac{2 (4)}{6}}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x^{a} = x^{\frac{2 \cdot 4}{2 \cdot 3}}$

Cancel the common factor.

$x^{a} = x^{\frac{2 \cdot 4}{2 \cdot 3}}$

Rewrite the expression.

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

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = 1.\overline{3}$

Mixed Number Form:

$a = 1 \frac{1}{3}$

Solve for a ( square root of x^3)/(x^(1/6))=x^a

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