Basic Math Examples

$(\sqrt{27 a^{3}}) (\sqrt{3 a^{3}}) (\sqrt{2 a})$

Step 1

Rewrite $27 a^{3}$ as ${(3 a)}^{2} \cdot (3 a)$ .

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Step 1.1

Factor $9$ out of $27$ .

$\sqrt{9 (3) a^{3}} \sqrt{3 a^{3}} \sqrt{2 a}$

Step 1.2

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

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

Step 1.3

Factor out $a^{2}$ .

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

Step 1.4

Move $3$ .

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

Step 1.5

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

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

Step 1.6

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor out $a^{2}$ .

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

Step 3.2

Reorder $3$ and $a^{2}$ .

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

Step 3.3

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $a$ .

$3 (a \cdot a) \sqrt{3 a} \sqrt{3 a} \sqrt{2 a}$

Step 5.2

Multiply $a$ by $a$ .

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

Step 6

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

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Step 6.1

Raise $\sqrt{3 a}$ to the power of $1$ .

$3 a^{2} ({\sqrt{3 a}}^{1} \sqrt{3 a}) \sqrt{2 a}$

Step 6.2

Raise $\sqrt{3 a}$ to the power of $1$ .

$3 a^{2} ({\sqrt{3 a}}^{1} {\sqrt{3 a}}^{1}) \sqrt{2 a}$

Step 6.3

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

$3 a^{2} {\sqrt{3 a}}^{1 + 1} \sqrt{2 a}$

Step 6.4

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

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

$3 a^{2} {({(3 a)}^{\frac{1}{2}})}^{2} \sqrt{2 a}$

Step 7.2

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

$3 a^{2} {(3 a)}^{\frac{1}{2} \cdot 2} \sqrt{2 a}$

Step 7.3

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

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

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

$3 a^{2} {(3 a)}^{1} \sqrt{2 a}$

Step 7.5

Simplify.

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

Step 8

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

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Step 8.1

Move $a$ .

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

Step 8.2

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

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Step 8.2.1

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

$3 (a^{1} a^{2}) \cdot 3 \sqrt{2 a}$

Step 8.2.2

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

$3 a^{1 + 2} \cdot 3 \sqrt{2 a}$

Step 8.3

Add $1$ and $2$ .

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

Step 9

Multiply $3$ by $3$ .

$9 a^{3} \sqrt{2 a}$