Solve for a ( sixth root of x)/((x^5)^(1/2))=x^a

$\frac{\sqrt[6]{x}}{{(x^{5})}^{\frac{1}{2}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[6]{x}}{{(x^{5})}^{\frac{1}{2}}}$ .

$x^{a} = \frac{\sqrt[6]{x}}{{(x^{5})}^{\frac{1}{2}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$x^{a} = \frac{x^{\frac{1}{6}}}{{(x^{5})}^{\frac{1}{2}}}$

Step 3

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

$x^{a} = \frac{x^{\frac{1}{6}}}{x^{5 (\frac{1}{2})}}$

Step 4

Move $x^{5 (\frac{1}{2})}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$x^{a} = x^{\frac{1}{6}} x^{- 5 (\frac{1}{2})}$

Step 5

Multiply $x^{\frac{1}{6}}$ by $x^{- 5 (\frac{1}{2})}$ by adding the exponents.

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

$x^{a} = x^{\frac{1}{6} - 5 (\frac{1}{2})}$

Simplify each term.

Combine $- 5$ and $\frac{1}{2}$ .

$x^{a} = x^{\frac{1}{6} + \frac{- 5}{2}}$

Move the negative in front of the fraction.

$x^{a} = x^{\frac{1}{6} - \frac{5}{2}}$

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{a} = x^{\frac{1}{6} - \frac{5}{2} \cdot \frac{3}{3}}$

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

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$x^{a} = x^{\frac{1}{6} - \frac{5 \cdot 3}{2 \cdot 3}}$

Multiply $2$ by $3$ .

$x^{a} = x^{\frac{1}{6} - \frac{5 \cdot 3}{6}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{1 - 5 \cdot 3}{6}}$

Simplify the numerator.

Multiply $- 5$ by $3$ .

$x^{a} = x^{\frac{1 - 15}{6}}$

Subtract $15$ from $1$ .

$x^{a} = x^{\frac{- 14}{6}}$

Cancel the common factor of $- 14$ and $6$ .

Factor $2$ out of $- 14$ .

$x^{a} = x^{\frac{2 (- 7)}{6}}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x^{a} = x^{\frac{2 \cdot - 7}{2 \cdot 3}}$

Cancel the common factor.

$x^{a} = x^{\frac{2 \cdot - 7}{2 \cdot 3}}$

Rewrite the expression.

$x^{a} = x^{\frac{- 7}{3}}$

Move the negative in front of the fraction.

$x^{a} = x^{- \frac{7}{3}}$

Step 6

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

$a = - \frac{7}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{7}{3}$

Decimal Form:

$a = - 2.\overline{3}$

Mixed Number Form:

$a = - 2 \frac{1}{3}$

Solve for a ( sixth root of x)/((x^5)^(1/2))=x^a

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