Algebra Examples

$\frac{\sqrt{5 k^{4}} + 3 \sqrt{2 k}}{\sqrt{3 k^{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 k^{4}}$ as ${(5 k^{4})}^{\frac{1}{2}}$ .

$\frac{{(5 k^{4})}^{\frac{1}{2}} + 3 \sqrt{2 k}}{\sqrt{3 k^{3}}}$

Step 1.2

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

$\frac{{(5 k^{4})}^{\frac{1}{2}} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.3

Apply the product rule to $5 k^{4}$ .

$\frac{5^{\frac{1}{2}} {(k^{4})}^{\frac{1}{2}} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.4

Multiply the exponents in ${(k^{4})}^{\frac{1}{2}}$ .

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Step 1.4.1

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

$\frac{5^{\frac{1}{2}} k^{4 (\frac{1}{2})} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $4$ .

$\frac{5^{\frac{1}{2}} k^{2 (2) \frac{1}{2}} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.4.2.2

Cancel the common factor.

$\frac{5^{\frac{1}{2}} k^{2 \cdot 2 \frac{1}{2}} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.4.2.3

Rewrite the expression.

$\frac{5^{\frac{1}{2}} k^{2} + 3 {(2 k)}^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.5

Apply the product rule to $2 k$ .

$\frac{5^{\frac{1}{2}} k^{2} + 3 (2^{\frac{1}{2}} k^{\frac{1}{2}})}{\sqrt{3 k^{3}}}$

Step 1.6

Factor $k^{\frac{1}{2}}$ out of $5^{\frac{1}{2}} k^{2} + 3 \cdot 2^{\frac{1}{2}} k^{\frac{1}{2}}$ .

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Step 1.6.1

Factor $k^{\frac{1}{2}}$ out of $5^{\frac{1}{2}} k^{2}$ .

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}}) + 3 \cdot 2^{\frac{1}{2}} k^{\frac{1}{2}}}{\sqrt{3 k^{3}}}$

Step 1.6.2

Factor $k^{\frac{1}{2}}$ out of $3 \cdot 2^{\frac{1}{2}} k^{\frac{1}{2}}$ .

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}}) + k^{\frac{1}{2}} (3 \cdot 2^{\frac{1}{2}})}{\sqrt{3 k^{3}}}$

Step 1.6.3

Factor $k^{\frac{1}{2}}$ out of $k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}}) + k^{\frac{1}{2}} (3 \cdot 2^{\frac{1}{2}})$ .

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{\sqrt{3 k^{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $3 k^{3}$ as $k^{2} \cdot (3 k)$ .

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Step 2.1.1

Factor out $k^{2}$ .

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{\sqrt{3 (k^{2} k)}}$

Step 2.1.2

Reorder $3$ and $k^{2}$ .

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{\sqrt{k^{2} \cdot 3 k}}$

Step 2.1.3

Add parentheses.

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{\sqrt{k^{2} \cdot (3 k)}}$

Step 2.2

Pull terms out from under the radical.

$\frac{k^{\frac{1}{2}} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{k \sqrt{3 k}}$

Step 3

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

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k \sqrt{3 k} k^{- \frac{1}{2}}}$

Step 4

Multiply $k$ by $k^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.1

Move $k^{- \frac{1}{2}}$ .

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{- \frac{1}{2}} k \sqrt{3 k}}$

Step 4.2

Multiply $k^{- \frac{1}{2}}$ by $k$ .

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Step 4.2.1

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

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{- \frac{1}{2}} k^{1} \sqrt{3 k}}$

Step 4.2.2

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

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{- \frac{1}{2} + 1} \sqrt{3 k}}$

Step 4.3

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

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{- \frac{1}{2} + \frac{2}{2}} \sqrt{3 k}}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{\frac{- 1 + 2}{2}} \sqrt{3 k}}$

Step 4.5

Add $- 1$ and $2$ .

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{\frac{1}{2}} \sqrt{3 k}}$

Step 5

Multiply $\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{\frac{1}{2}} \sqrt{3 k}}$ by $\frac{\sqrt{3 k}}{\sqrt{3 k}}$ .

$\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{\frac{1}{2}} \sqrt{3 k}} \cdot \frac{\sqrt{3 k}}{\sqrt{3 k}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}}{k^{\frac{1}{2}} \sqrt{3 k}}$ by $\frac{\sqrt{3 k}}{\sqrt{3 k}}$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} \sqrt{3 k} \sqrt{3 k}}$

Step 6.2

Move $\sqrt{3 k}$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} (\sqrt{3 k} \sqrt{3 k})}$

Step 6.3

Raise $\sqrt{3 k}$ to the power of $1$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} ({\sqrt{3 k}}^{1} \sqrt{3 k})}$

Step 6.4

Raise $\sqrt{3 k}$ to the power of $1$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} ({\sqrt{3 k}}^{1} {\sqrt{3 k}}^{1})}$

Step 6.5

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {\sqrt{3 k}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {\sqrt{3 k}}^{2}}$

Step 6.7

Rewrite ${\sqrt{3 k}}^{2}$ as $3 k$ .

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Step 6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 k}$ as ${(3 k)}^{\frac{1}{2}}$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {({(3 k)}^{\frac{1}{2}})}^{2}}$

Step 6.7.2

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {(3 k)}^{\frac{1}{2} \cdot 2}}$

Step 6.7.3

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {(3 k)}^{\frac{2}{2}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {(3 k)}^{\frac{2}{2}}}$

Step 6.7.4.2

Rewrite the expression.

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} {(3 k)}^{1}}$

Step 6.7.5

Simplify.

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{1}{2}} (3 k)}$

Step 7

Multiply $k^{\frac{1}{2}}$ by $k$ by adding the exponents.

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Step 7.1

Move $k$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k \cdot k^{\frac{1}{2}} \cdot 3}$

Step 7.2

Multiply $k$ by $k^{\frac{1}{2}}$ .

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Step 7.2.1

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{1} k^{\frac{1}{2}} \cdot 3}$

Step 7.2.2

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{1 + \frac{1}{2}} \cdot 3}$

Step 7.3

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

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{2}{2} + \frac{1}{2}} \cdot 3}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{2 + 1}{2}} \cdot 3}$

Step 7.5

Add $2$ and $1$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{k^{\frac{3}{2}} \cdot 3}$

Step 8

Move $3$ to the left of $k^{\frac{3}{2}}$ .

$\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{3 \cdot k^{\frac{3}{2}}}$

Step 9

Reorder factors in $\frac{(5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}}) \sqrt{3 k}}{3 k^{\frac{3}{2}}}$ .

$\frac{\sqrt{3 k} (5^{\frac{1}{2}} k^{\frac{3}{2}} + 3 \cdot 2^{\frac{1}{2}})}{3 k^{\frac{3}{2}}}$