Solve for a x/( square root of x^5)=x^a

$\frac{x}{\sqrt{x^{5}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{x}{\sqrt{x^{5}}}$ .

$x^{a} = \frac{x}{\sqrt{x^{5}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$x^{a} = \frac{x}{x^{\frac{5}{2}}}$

Step 3

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

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

Step 4

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

Multiply $x$ by $x^{- \frac{5}{2}}$ .

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Subtract $5$ from $2$ .

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

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = - 1.5$

Mixed Number Form:

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