Basic Math Examples

$(5 - \sqrt{10 a}) (3 - \sqrt{35 a})$

Step 1

Expand $(5 - \sqrt{10 a}) (3 - \sqrt{35 a})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$5 (3 - \sqrt{35 a}) - \sqrt{10 a} (3 - \sqrt{35 a})$

Step 1.2

Apply the distributive property.

$5 \cdot 3 + 5 (- \sqrt{35 a}) - \sqrt{10 a} (3 - \sqrt{35 a})$

Step 1.3

Apply the distributive property.

$5 \cdot 3 + 5 (- \sqrt{35 a}) - \sqrt{10 a} \cdot 3 - \sqrt{10 a} (- \sqrt{35 a})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $5$ by $3$ .

$15 + 5 (- \sqrt{35 a}) - \sqrt{10 a} \cdot 3 - \sqrt{10 a} (- \sqrt{35 a})$

Step 2.1.2

Multiply $- 1$ by $5$ .

$15 - 5 \sqrt{35 a} - \sqrt{10 a} \cdot 3 - \sqrt{10 a} (- \sqrt{35 a})$

Step 2.1.3

Multiply $3$ by $- 1$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} - \sqrt{10 a} (- \sqrt{35 a})$

Step 2.1.4

Multiply $- \sqrt{10 a} (- \sqrt{35 a})$ .

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + 1 \sqrt{10 a} \sqrt{35 a}$

Step 2.1.4.2

Multiply $\sqrt{10 a}$ by $1$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{10 a} \sqrt{35 a}$

Step 2.1.4.3

Combine using the product rule for radicals.

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{10 a (35 a)}$

Step 2.1.4.4

Multiply $35$ by $10$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{350 a \cdot a}$

Step 2.1.4.5

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

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{350 (a^{1} a)}$

Step 2.1.4.6

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

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{350 (a^{1} a^{1})}$

Step 2.1.4.7

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

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{350 a^{1 + 1}}$

Step 2.1.4.8

Add $1$ and $1$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{350 a^{2}}$

Step 2.1.5

Rewrite $350 a^{2}$ as ${(5 a)}^{2} \cdot 14$ .

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Step 2.1.5.1

Factor $25$ out of $350$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{25 (14) a^{2}}$

Step 2.1.5.2

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

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{5^{2} \cdot 14 a^{2}}$

Step 2.1.5.3

Move $14$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{5^{2} a^{2} \cdot 14}$

Step 2.1.5.4

Rewrite $5^{2} a^{2}$ as ${(5 a)}^{2}$ .

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + \sqrt{{(5 a)}^{2} \cdot 14}$

Step 2.1.6

Pull terms out from under the radical.

$15 - 5 \sqrt{35 a} - 3 \sqrt{10 a} + 5 a \sqrt{14}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $- 3 \sqrt{10 a}$ .

$15 - 5 \sqrt{35 a} + 5 a \sqrt{14} - 3 \sqrt{10 a}$

Step 2.2.2

Move $- 5 \sqrt{35 a}$ .

$15 + 5 a \sqrt{14} - 5 \sqrt{35 a} - 3 \sqrt{10 a}$

Step 2.2.3

Reorder $15$ and $5 a \sqrt{14}$ .

$5 a \sqrt{14} + 15 - 5 \sqrt{35 a} - 3 \sqrt{10 a}$