Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 5$ by $2$ .

$x^{a} = x^{\frac{5 - 10}{2}}$

Step 4.5.2

Subtract $10$ from $5$ .

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

Step 4.6

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$a = - 2.5$

Mixed Number Form:

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