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

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

Simplify each term.

Multiply $- \frac{1}{2} \cdot 5$ .

Multiply $5$ by $- 1$ .

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

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

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

Move the negative in front of the fraction.

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

Combine the numerators over the common denominator.

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

Subtract $5$ from $1$ .

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

Divide $- 4$ by $2$ .

$x^{a} = x^{- 2}$

Step 6

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

$a = - 2$