Basic Math Examples

$\sqrt{5 a^{4}} \sqrt{20 a^{3}}$

Step 1

Rewrite $5 a^{4}$ as ${(a^{2})}^{2} \cdot 5$ .

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Step 1.1

Rewrite $a^{4}$ as ${(a^{2})}^{2}$ .

$\sqrt{5 {(a^{2})}^{2}} \sqrt{20 a^{3}}$

Step 1.2

Reorder $5$ and ${(a^{2})}^{2}$ .

$\sqrt{{(a^{2})}^{2} \cdot 5} \sqrt{20 a^{3}}$

$\sqrt{{(a^{2})}^{2} \cdot 5} \sqrt{20 a^{3}}$

Step 2

Pull terms out from under the radical.

$a^{2} \sqrt{5} \sqrt{20 a^{3}}$

Step 3

Rewrite $20 a^{3}$ as ${(2 a)}^{2} \cdot (5 a)$ .

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Step 3.1

Factor $4$ out of $20$ .

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

Step 3.2

Rewrite $4$ as $2^{2}$ .

$a^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 a^{3}}$

Step 3.3

Factor out $a^{2}$ .

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

Step 3.4

Move $5$ .

$a^{2} \sqrt{5} \sqrt{2^{2} a^{2} \cdot 5 a}$

Step 3.5

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

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

Step 3.6

Add parentheses.

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 5.1

Move $a$ .

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

Step 5.2

Multiply $a$ by $a^{2}$ .

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Step 5.2.1

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

$a^{1} a^{2} \sqrt{5} (2 \sqrt{5 a})$

Step 5.2.2

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

$a^{1 + 2} \sqrt{5} (2 \sqrt{5 a})$

$a^{1 + 2} \sqrt{5} (2 \sqrt{5 a})$

Step 5.3

Add $1$ and $2$ .

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

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

Step 6

Multiply $a^{3} \sqrt{5} (2 \sqrt{5 a})$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $5$ by $5$ .

$a^{3} (2 \sqrt{25 a})$

$a^{3} (2 \sqrt{25 a})$

Step 7

Simplify the expression.

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Step 7.1

Rewrite using the commutative property of multiplication.

$2 a^{3} \sqrt{25 a}$

Step 7.2

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

$2 a^{3} \sqrt{5^{2} a}$

$2 a^{3} \sqrt{5^{2} a}$

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $5$ by $2$ .

$10 a^{3} \sqrt{a}$

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