Algebra Examples

$r = \sqrt{\frac{a}{π}}$

Step 1

Simplify $\sqrt{\frac{a}{π}}$ .

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Step 1.1

Rewrite $\sqrt{\frac{a}{π}}$ as $\frac{\sqrt{a}}{\sqrt{π}}$ .

$r = \frac{\sqrt{a}}{\sqrt{π}}$

Step 1.2

Multiply $\frac{\sqrt{a}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

$r = \frac{\sqrt{a}}{\sqrt{π}} \cdot \frac{\sqrt{π}}{\sqrt{π}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{a}}{\sqrt{π}}$ by $\frac{\sqrt{π}}{\sqrt{π}}$ .

$r = \frac{\sqrt{a} \sqrt{π}}{\sqrt{π} \sqrt{π}}$

Step 1.3.2

Raise $\sqrt{π}$ to the power of $1$ .

$r = \frac{\sqrt{a} \sqrt{π}}{{\sqrt{π}}^{1} \sqrt{π}}$

Step 1.3.3

Raise $\sqrt{π}$ to the power of $1$ .

$r = \frac{\sqrt{a} \sqrt{π}}{{\sqrt{π}}^{1} {\sqrt{π}}^{1}}$

Step 1.3.4

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

$r = \frac{\sqrt{a} \sqrt{π}}{{\sqrt{π}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$r = \frac{\sqrt{a} \sqrt{π}}{{\sqrt{π}}^{2}}$

Step 1.3.6

Rewrite ${\sqrt{π}}^{2}$ as $π$ .

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Step 1.3.6.1

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

$r = \frac{\sqrt{a} \sqrt{π}}{{(π^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$r = \frac{\sqrt{a} \sqrt{π}}{π^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$r = \frac{\sqrt{a} \sqrt{π}}{π^{\frac{2}{2}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$r = \frac{\sqrt{a} \sqrt{π}}{π^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$r = \frac{\sqrt{a} \sqrt{π}}{π^{1}}$

Step 1.3.6.5

Simplify.

$r = \frac{\sqrt{a} \sqrt{π}}{π}$

Step 1.4

Combine using the product rule for radicals.

$r = \frac{\sqrt{a π}}{π}$

Step 2

Set the radicand in $\sqrt{a π}$ greater than or equal to $0$ to find where the expression is defined.

$a π \geq 0$

Step 3

Divide each term in $a π \geq 0$ by $π$ and simplify.

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Step 3.1

Divide each term in $a π \geq 0$ by $π$ .

$\frac{a π}{π} \geq \frac{0}{π}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $π$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{a π}{π} \geq \frac{0}{π}$

Step 3.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{0}{π}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $π$ .

$a \geq 0$

Step 4

The domain is all values of $a$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${a | a \geq 0}$

Step 5