Algebra Examples

$\frac{\sqrt{z}}{\sqrt{z} - \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{z}}{\sqrt{z} - \sqrt{3}}$ by $\frac{\sqrt{z} + \sqrt{3}}{\sqrt{z} + \sqrt{3}}$ .

$\frac{\sqrt{z}}{\sqrt{z} - \sqrt{3}} \cdot \frac{\sqrt{z} + \sqrt{3}}{\sqrt{z} + \sqrt{3}}$

Step 2

Multiply $\frac{\sqrt{z}}{\sqrt{z} - \sqrt{3}}$ by $\frac{\sqrt{z} + \sqrt{3}}{\sqrt{z} + \sqrt{3}}$ .

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{(\sqrt{z} - \sqrt{3}) (\sqrt{z} + \sqrt{3})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{{\sqrt{z}}^{2} + \sqrt{z \cdot 3} - \sqrt{z \cdot 3} - {\sqrt{3}}^{2}}$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{{(z^{\frac{1}{2}})}^{2} - {\sqrt{3}}^{2}}$

Step 4.1.2

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z^{\frac{1}{2} \cdot 2} - {\sqrt{3}}^{2}}$

Step 4.1.3

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z^{\frac{2}{2}} - {\sqrt{3}}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z^{\frac{2}{2}} - {\sqrt{3}}^{2}}$

Step 4.1.4.2

Rewrite the expression.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z^{1} - {\sqrt{3}}^{2}}$

Step 4.1.5

Simplify.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - {\sqrt{3}}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - {(3^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 3^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 3^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 3^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 3^{1}}$

Step 4.2.5

Evaluate the exponent.

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 1 \cdot 3}$

Step 4.3

Multiply $- 1$ by $3$ .

$\frac{\sqrt{z} (\sqrt{z} + \sqrt{3})}{z - 3}$

Step 5

Apply the distributive property.

$\frac{\sqrt{z} \sqrt{z} + \sqrt{z} \sqrt{3}}{z - 3}$

Step 6

Multiply $\sqrt{z} \sqrt{z}$ .

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Step 6.1

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

$\frac{{\sqrt{z}}^{1} \sqrt{z} + \sqrt{z} \sqrt{3}}{z - 3}$

Step 6.2

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

$\frac{{\sqrt{z}}^{1} {\sqrt{z}}^{1} + \sqrt{z} \sqrt{3}}{z - 3}$

Step 6.3

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

$\frac{{\sqrt{z}}^{1 + 1} + \sqrt{z} \sqrt{3}}{z - 3}$

Step 6.4

Add $1$ and $1$ .

$\frac{{\sqrt{z}}^{2} + \sqrt{z} \sqrt{3}}{z - 3}$

Step 7

Combine using the product rule for radicals.

$\frac{{\sqrt{z}}^{2} + \sqrt{z \cdot 3}}{z - 3}$

Step 8

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

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Step 8.1

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

$\frac{{(z^{\frac{1}{2}})}^{2} + \sqrt{z \cdot 3}}{z - 3}$

Step 8.2

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

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

Step 8.3

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

$\frac{z^{\frac{2}{2}} + \sqrt{z \cdot 3}}{z - 3}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$\frac{z^{\frac{2}{2}} + \sqrt{z \cdot 3}}{z - 3}$

Step 8.4.2

Rewrite the expression.

$\frac{z^{1} + \sqrt{z \cdot 3}}{z - 3}$

Step 8.5

Simplify.

$\frac{z + \sqrt{z \cdot 3}}{z - 3}$

Step 9

Reorder factors in $z + \sqrt{z \cdot 3}$ .

$\frac{z + \sqrt{3 z}}{z - 3}$

Step 10

Simplify the numerator.

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Step 10.1

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

$\frac{z + {(3 z)}^{\frac{1}{2}}}{z - 3}$

Step 10.2

Apply the product rule to $3 z$ .

$\frac{z + 3^{\frac{1}{2}} z^{\frac{1}{2}}}{z - 3}$

Step 10.3

Factor $z^{\frac{1}{2}}$ out of $z + 3^{\frac{1}{2}} z^{\frac{1}{2}}$ .

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Step 10.3.1

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

$\frac{z^{1} + 3^{\frac{1}{2}} z^{\frac{1}{2}}}{z - 3}$

Step 10.3.2

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

$\frac{z^{\frac{1}{2}} z^{\frac{1}{2}} + 3^{\frac{1}{2}} z^{\frac{1}{2}}}{z - 3}$

Step 10.3.3

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

$\frac{z^{\frac{1}{2}} z^{\frac{1}{2}} + z^{\frac{1}{2}} \cdot 3^{\frac{1}{2}}}{z - 3}$

Step 10.3.4

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

$\frac{z^{\frac{1}{2}} (z^{\frac{1}{2}} + 3^{\frac{1}{2}})}{z - 3}$