Basic Math Examples

$\frac{\sqrt{a} + \sqrt{z}}{2 \sqrt{a} + \sqrt{z}}$

Step 1

Multiply $\frac{\sqrt{a} + \sqrt{z}}{2 \sqrt{a} + \sqrt{z}}$ by $\frac{2 \sqrt{a} - \sqrt{z}}{2 \sqrt{a} - \sqrt{z}}$ .

$\frac{\sqrt{a} + \sqrt{z}}{2 \sqrt{a} + \sqrt{z}} \cdot \frac{2 \sqrt{a} - \sqrt{z}}{2 \sqrt{a} - \sqrt{z}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{a} + \sqrt{z}}{2 \sqrt{a} + \sqrt{z}}$ by $\frac{2 \sqrt{a} - \sqrt{z}}{2 \sqrt{a} - \sqrt{z}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 {\sqrt{a}}^{2} - 2 \sqrt{z a} + 2 \sqrt{z a} - {\sqrt{z}}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

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

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Step 2.3.1.1

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 {(a^{\frac{1}{2}})}^{2} - {\sqrt{z}}^{2}}$

Step 2.3.1.2

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a^{\frac{1}{2} \cdot 2} - {\sqrt{z}}^{2}}$

Step 2.3.1.3

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a^{\frac{2}{2}} - {\sqrt{z}}^{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a^{\frac{2}{2}} - {\sqrt{z}}^{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a^{1} - {\sqrt{z}}^{2}}$

Step 2.3.1.5

Simplify.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - {\sqrt{z}}^{2}}$

Step 2.3.2

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

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Step 2.3.2.1

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - {(z^{\frac{1}{2}})}^{2}}$

Step 2.3.2.2

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - z^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.3

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

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - z^{\frac{2}{2}}}$

Step 2.3.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.4.1

Cancel the common factor.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - z^{\frac{2}{2}}}$

Step 2.3.2.4.2

Rewrite the expression.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - z^{1}}$

Step 2.3.2.5

Simplify.

$\frac{(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})}{4 a - z}$

Step 3

Expand $(\sqrt{a} + \sqrt{z}) (2 \sqrt{a} - \sqrt{z})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a} - \sqrt{z}) + \sqrt{z} (2 \sqrt{a} - \sqrt{z})}{4 a - z}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a}) + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a} - \sqrt{z})}{4 a - z}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a}) + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \sqrt{a} \sqrt{a} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.2

Multiply $2 \sqrt{a} \sqrt{a}$ .

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Step 4.1.2.1

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

$\frac{2 ({\sqrt{a}}^{1} \sqrt{a}) + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.2.2

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

$\frac{2 ({\sqrt{a}}^{1} {\sqrt{a}}^{1}) + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.2.3

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

$\frac{2 {\sqrt{a}}^{1 + 1} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{a}}^{2} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{2 {(a^{\frac{1}{2}})}^{2} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3.2

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

$\frac{2 a^{\frac{1}{2} \cdot 2} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3.3

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

$\frac{2 a^{\frac{2}{2}} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{2 a^{\frac{2}{2}} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{2 a^{1} + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.3.5

Simplify.

$\frac{2 a + \sqrt{a} (- \sqrt{z}) + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 a - \sqrt{a} \sqrt{z} + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.5

Combine using the product rule for radicals.

$\frac{2 a - \sqrt{z a} + \sqrt{z} (2 \sqrt{a}) + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.6

Rewrite using the commutative property of multiplication.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{z} \sqrt{a} + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.7

Combine using the product rule for radicals.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} + \sqrt{z} (- \sqrt{z})}{4 a - z}$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - \sqrt{z} \sqrt{z}}{4 a - z}$

Step 4.1.9

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

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Step 4.1.9.1

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - ({\sqrt{z}}^{1} \sqrt{z})}{4 a - z}$

Step 4.1.9.2

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - ({\sqrt{z}}^{1} {\sqrt{z}}^{1})}{4 a - z}$

Step 4.1.9.3

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - {\sqrt{z}}^{1 + 1}}{4 a - z}$

Step 4.1.9.4

Add $1$ and $1$ .

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - {\sqrt{z}}^{2}}{4 a - z}$

Step 4.1.10

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

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Step 4.1.10.1

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - {(z^{\frac{1}{2}})}^{2}}{4 a - z}$

Step 4.1.10.2

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - z^{\frac{1}{2} \cdot 2}}{4 a - z}$

Step 4.1.10.3

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

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - z^{\frac{2}{2}}}{4 a - z}$

Step 4.1.10.4

Cancel the common factor of $2$ .

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Step 4.1.10.4.1

Cancel the common factor.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - z^{\frac{2}{2}}}{4 a - z}$

Step 4.1.10.4.2

Rewrite the expression.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - z^{1}}{4 a - z}$

Step 4.1.10.5

Simplify.

$\frac{2 a - \sqrt{z a} + 2 \sqrt{a z} - z}{4 a - z}$

Step 4.2

Add $- \sqrt{z a}$ and $2 \sqrt{a z}$ .

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Step 4.2.1

Reorder $z$ and $a$ .

$\frac{2 a - \sqrt{a z} + 2 \sqrt{a z} - z}{4 a - z}$

Step 4.2.2

Add $- \sqrt{a z}$ and $2 \sqrt{a z}$ .

$\frac{2 a + \sqrt{a z} - z}{4 a - z}$