Linear Algebra Examples

$w = \sqrt{\frac{k}{m}}$

Step 1

Simplify $\sqrt{\frac{k}{m}}$ .

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Step 1.1

Rewrite $\sqrt{\frac{k}{m}}$ as $\frac{\sqrt{k}}{\sqrt{m}}$ .

$w = \frac{\sqrt{k}}{\sqrt{m}}$

Step 1.2

Multiply $\frac{\sqrt{k}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$w = \frac{\sqrt{k}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{k}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$w = \frac{\sqrt{k} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

Step 1.3.2

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

$w = \frac{\sqrt{k} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

Step 1.3.3

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

$w = \frac{\sqrt{k} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

Step 1.3.4

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

$w = \frac{\sqrt{k} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$w = \frac{\sqrt{k} \sqrt{m}}{{\sqrt{m}}^{2}}$

Step 1.3.6

Rewrite ${\sqrt{m}}^{2}$ as $m$ .

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Step 1.3.6.1

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

$w = \frac{\sqrt{k} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$w = \frac{\sqrt{k} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$w = \frac{\sqrt{k} \sqrt{m}}{m^{\frac{2}{2}}}$

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.

$w = \frac{\sqrt{k} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$w = \frac{\sqrt{k} \sqrt{m}}{m^{1}}$

Step 1.3.6.5

Simplify.

$w = \frac{\sqrt{k} \sqrt{m}}{m}$

Step 1.4

Combine using the product rule for radicals.

$w = \frac{\sqrt{k m}}{m}$

Step 2

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

$k m \geq 0$

Step 3

Divide each term in $k m \geq 0$ by $m$ and simplify.

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Step 3.1

Divide each term in $k m \geq 0$ by $m$ .

$\frac{k m}{m} \geq \frac{0}{m}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $m$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{k m}{m} \geq \frac{0}{m}$

Step 3.2.1.2

Divide $k$ by $1$ .

$k \geq \frac{0}{m}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $m$ .

$k \geq 0$

Step 4

Set the denominator in $\frac{\sqrt{k m}}{m}$ equal to $0$ to find where the expression is undefined.

$m = 0$

Step 5

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${k | k > 0}$