Basic Math Examples

$\frac{\sqrt{2 a}}{\sqrt{5 a} - \sqrt{3 b}}$

Step 1

Multiply $\frac{\sqrt{2 a}}{\sqrt{5 a} - \sqrt{3 b}}$ by $\frac{\sqrt{5 a} + \sqrt{3 b}}{\sqrt{5 a} + \sqrt{3 b}}$ .

$\frac{\sqrt{2 a}}{\sqrt{5 a} - \sqrt{3 b}} \cdot \frac{\sqrt{5 a} + \sqrt{3 b}}{\sqrt{5 a} + \sqrt{3 b}}$

Step 2

Multiply $\frac{\sqrt{2 a}}{\sqrt{5 a} - \sqrt{3 b}}$ by $\frac{\sqrt{5 a} + \sqrt{3 b}}{\sqrt{5 a} + \sqrt{3 b}}$ .

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{(\sqrt{5 a} - \sqrt{3 b}) (\sqrt{5 a} + \sqrt{3 b})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{\sqrt{5 a}}^{2} + \sqrt{15 a b} - \sqrt{15 a b} - {\sqrt{3 b}}^{2}}$

Step 4

Simplify.

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Step 4.1

Rewrite ${\sqrt{5 a}}^{2}$ as $5 a$ .

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Step 4.1.1

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{({(5 a)}^{\frac{1}{2}})}^{2} - {\sqrt{3 b}}^{2}}$

Step 4.1.2

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{(5 a)}^{\frac{1}{2} \cdot 2} - {\sqrt{3 b}}^{2}}$

Step 4.1.3

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{(5 a)}^{\frac{2}{2}} - {\sqrt{3 b}}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{(5 a)}^{\frac{2}{2}} - {\sqrt{3 b}}^{2}}$

Step 4.1.4.2

Rewrite the expression.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{{(5 a)}^{1} - {\sqrt{3 b}}^{2}}$

Step 4.1.5

Simplify.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {\sqrt{3 b}}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {({(3 b)}^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {(3 b)}^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {(3 b)}^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {(3 b)}^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - {(3 b)}^{1}}$

Step 4.2.5

Simplify.

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - (3 b)}$

Step 4.3

Multiply $3$ by $- 1$ .

$\frac{\sqrt{2 a} (\sqrt{5 a} + \sqrt{3 b})}{5 a - 3 b}$

Step 5

Apply the distributive property.

$\frac{\sqrt{2 a} \sqrt{5 a} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6

Multiply $\sqrt{2 a} \sqrt{5 a}$ .

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 a (5 a)} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6.2

Multiply $5$ by $2$ .

$\frac{\sqrt{10 a \cdot a} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6.3

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

$\frac{\sqrt{10 (a^{1} a)} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6.4

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

$\frac{\sqrt{10 (a^{1} a^{1})} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6.5

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

$\frac{\sqrt{10 a^{1 + 1}} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 6.6

Add $1$ and $1$ .

$\frac{\sqrt{10 a^{2}} + \sqrt{2 a} \sqrt{3 b}}{5 a - 3 b}$

Step 7

Multiply $\sqrt{2 a} \sqrt{3 b}$ .

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 a^{2}} + \sqrt{2 a (3 b)}}{5 a - 3 b}$

Step 7.2

Multiply $3$ by $2$ .

$\frac{\sqrt{10 a^{2}} + \sqrt{6 a b}}{5 a - 3 b}$

Step 8

Simplify each term.

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Step 8.1

Reorder $10$ and $a^{2}$ .

$\frac{\sqrt{a^{2} \cdot 10} + \sqrt{6 a b}}{5 a - 3 b}$

Step 8.2

Pull terms out from under the radical.

$\frac{a \sqrt{10} + \sqrt{6 a b}}{5 a - 3 b}$

Step 9

Simplify the numerator.

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Step 9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{a \cdot 10^{\frac{1}{2}} + \sqrt{6 a b}}{5 a - 3 b}$

Step 9.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 a b}$ as ${(6 a b)}^{\frac{1}{2}}$ .

$\frac{a \cdot 10^{\frac{1}{2}} + {(6 a b)}^{\frac{1}{2}}}{5 a - 3 b}$

Step 9.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.3.1

Apply the product rule to $6 a b$ .

$\frac{a \cdot 10^{\frac{1}{2}} + {(6 a)}^{\frac{1}{2}} b^{\frac{1}{2}}}{5 a - 3 b}$

Step 9.3.2

Apply the product rule to $6 a$ .

$\frac{a \cdot 10^{\frac{1}{2}} + 6^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}}{5 a - 3 b}$

Step 9.4

Factor $a^{\frac{1}{2}}$ out of $a \cdot 10^{\frac{1}{2}} + 6^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

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Step 9.4.1

Reorder $a$ and $10^{\frac{1}{2}}$ .

$\frac{10^{\frac{1}{2}} \cdot a + 6^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}}{5 a - 3 b}$

Step 9.4.2

Factor $a^{\frac{1}{2}}$ out of $10^{\frac{1}{2}} \cdot a$ .

$\frac{a^{\frac{1}{2}} (10^{\frac{1}{2}} \cdot a^{\frac{1}{2}}) + 6^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}}{5 a - 3 b}$

Step 9.4.3

Factor $a^{\frac{1}{2}}$ out of $6^{\frac{1}{2}} a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} (10^{\frac{1}{2}} \cdot a^{\frac{1}{2}}) + a^{\frac{1}{2}} (6^{\frac{1}{2}} b^{\frac{1}{2}})}{5 a - 3 b}$

Step 9.4.4

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (10^{\frac{1}{2}} \cdot a^{\frac{1}{2}}) + a^{\frac{1}{2}} (6^{\frac{1}{2}} b^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} (10^{\frac{1}{2}} \cdot a^{\frac{1}{2}} + 6^{\frac{1}{2}} b^{\frac{1}{2}})}{5 a - 3 b}$

$\frac{a^{\frac{1}{2}} (10^{\frac{1}{2}} a^{\frac{1}{2}} + 6^{\frac{1}{2}} b^{\frac{1}{2}})}{5 a - 3 b}$