Algebra Examples

$(\sqrt{3 x} + \sqrt{5}) (\sqrt{15 x} + 2 \sqrt{30})$

Step 1

Expand $(\sqrt{3 x} + \sqrt{5}) (\sqrt{15 x} + 2 \sqrt{30})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{3 x} (\sqrt{15 x} + 2 \sqrt{30}) + \sqrt{5} (\sqrt{15 x} + 2 \sqrt{30})$

Step 1.2

Apply the distributive property.

$\sqrt{3 x} \sqrt{15 x} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} (\sqrt{15 x} + 2 \sqrt{30})$

Step 1.3

Apply the distributive property.

$\sqrt{3 x} \sqrt{15 x} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2

Simplify each term.

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Step 2.1

Multiply $\sqrt{3 x} \sqrt{15 x}$ .

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{3 x (15 x)} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.1.2

Multiply $15$ by $3$ .

$\sqrt{45 x \cdot x} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.1.3

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

$\sqrt{45 (x^{1} x)} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.1.4

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

$\sqrt{45 (x^{1} x^{1})} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.1.5

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

$\sqrt{45 x^{1 + 1}} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.1.6

Add $1$ and $1$ .

$\sqrt{45 x^{2}} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.2

Rewrite $45 x^{2}$ as ${(3 x)}^{2} \cdot 5$ .

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Step 2.2.1

Factor $9$ out of $45$ .

$\sqrt{9 (5) x^{2}} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.2.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 5 x^{2}} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.2.3

Move $5$ .

$\sqrt{3^{2} x^{2} \cdot 5} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.2.4

Rewrite $3^{2} x^{2}$ as ${(3 x)}^{2}$ .

$\sqrt{{(3 x)}^{2} \cdot 5} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.3

Pull terms out from under the radical.

$3 x \sqrt{5} + \sqrt{3 x} (2 \sqrt{30}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.4

Multiply $\sqrt{3 x} (2 \sqrt{30})$ .

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Step 2.4.1

Combine using the product rule for radicals.

$3 x \sqrt{5} + 2 \sqrt{3 x \cdot 30} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.4.2

Multiply $30$ by $3$ .

$3 x \sqrt{5} + 2 \sqrt{90 x} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.5

Rewrite $90 x$ as $3^{2} \cdot (10 x)$ .

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Step 2.5.1

Factor $9$ out of $90$ .

$3 x \sqrt{5} + 2 \sqrt{9 (10) x} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.5.2

Rewrite $9$ as $3^{2}$ .

$3 x \sqrt{5} + 2 \sqrt{3^{2} \cdot 10 x} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.5.3

Add parentheses.

$3 x \sqrt{5} + 2 \sqrt{3^{2} \cdot (10 x)} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.6

Pull terms out from under the radical.

$3 x \sqrt{5} + 2 (3 \sqrt{10 x}) + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.7

Multiply $3$ by $2$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{5} \sqrt{15 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.8

Multiply $\sqrt{5} \sqrt{15 x}$ .

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Step 2.8.1

Combine using the product rule for radicals.

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{5 (15 x)} + \sqrt{5} (2 \sqrt{30})$

Step 2.8.2

Multiply $5$ by $15$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{75 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.9

Rewrite $75 x$ as $5^{2} \cdot (3 x)$ .

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Step 2.9.1

Factor $25$ out of $75$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{25 (3) x} + \sqrt{5} (2 \sqrt{30})$

Step 2.9.2

Rewrite $25$ as $5^{2}$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{5^{2} \cdot 3 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.9.3

Add parentheses.

$3 x \sqrt{5} + 6 \sqrt{10 x} + \sqrt{5^{2} \cdot (3 x)} + \sqrt{5} (2 \sqrt{30})$

Step 2.10

Pull terms out from under the radical.

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + \sqrt{5} (2 \sqrt{30})$

Step 2.11

Multiply $\sqrt{5} (2 \sqrt{30})$ .

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Step 2.11.1

Combine using the product rule for radicals.

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 2 \sqrt{5 \cdot 30}$

Step 2.11.2

Multiply $5$ by $30$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 2 \sqrt{150}$

Step 2.12

Rewrite $150$ as $5^{2} \cdot 6$ .

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Step 2.12.1

Factor $25$ out of $150$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 2 \sqrt{25 (6)}$

Step 2.12.2

Rewrite $25$ as $5^{2}$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 2 \sqrt{5^{2} \cdot 6}$

Step 2.13

Pull terms out from under the radical.

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 2 (5 \sqrt{6})$

Step 2.14

Multiply $5$ by $2$ .

$3 x \sqrt{5} + 6 \sqrt{10 x} + 5 \sqrt{3 x} + 10 \sqrt{6}$