Basic Math Examples

$4 \sqrt{5 a} \cdot (3 \sqrt{10 a^{3}})$

Step 1

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

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Step 1.1

Factor out $a^{2}$ .

$4 \sqrt{5 a} \cdot (3 \sqrt{10 (a^{2} a)})$

Step 1.2

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

$4 \sqrt{5 a} \cdot (3 \sqrt{a^{2} \cdot 10 a})$

Step 1.3

Add parentheses.

$4 \sqrt{5 a} \cdot (3 \sqrt{a^{2} \cdot (10 a)})$

$4 \sqrt{5 a} \cdot (3 \sqrt{a^{2} \cdot (10 a)})$

Step 2

Pull terms out from under the radical.

$4 \sqrt{5 a} \cdot (3 (a \sqrt{10 a}))$

Step 3

Multiply $4 \sqrt{5 a} (3 a \sqrt{10 a})$ .

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Step 3.1

Multiply $3$ by $4$ .

$12 \sqrt{5 a} (a \sqrt{10 a})$

Step 3.2

Combine using the product rule for radicals.

$12 (a \sqrt{5 a (10 a)})$

Step 3.3

Multiply $10$ by $5$ .

$12 (a \sqrt{50 a \cdot a})$

Step 3.4

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

$12 (a \sqrt{50 (a^{1} a)})$

Step 3.5

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

$12 (a \sqrt{50 (a^{1} a^{1})})$

Step 3.6

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

$12 (a \sqrt{50 a^{1 + 1}})$

Step 3.7

Add $1$ and $1$ .

$12 (a \sqrt{50 a^{2}})$

$12 (a \sqrt{50 a^{2}})$

Step 4

Rewrite $50 a^{2}$ as ${(5 a)}^{2} \cdot 2$ .

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Step 4.1

Factor $25$ out of $50$ .

$12 (a \sqrt{25 (2) a^{2}})$

Step 4.2

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

$12 (a \sqrt{5^{2} \cdot 2 a^{2}})$

Step 4.3

Move $2$ .

$12 (a \sqrt{5^{2} a^{2} \cdot 2})$

Step 4.4

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

$12 (a \sqrt{{(5 a)}^{2} \cdot 2})$

$12 (a \sqrt{{(5 a)}^{2} \cdot 2})$

Step 5

Pull terms out from under the radical.

$12 (a (5 a \sqrt{2}))$

Step 6

Multiply $a$ by $a$ by adding the exponents.

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Step 6.1

Move $a$ .

$12 (a \cdot a (5 \sqrt{2}))$

Step 6.2

Multiply $a$ by $a$ .

$12 (a^{2} (5 \sqrt{2}))$

$12 (a^{2} (5 \sqrt{2}))$

Step 7

Move $5$ to the left of $a^{2}$ .

$12 (5 \cdot a^{2} \sqrt{2})$

Step 8

Multiply $5$ by $12$ .

$60 a^{2} \sqrt{2}$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$