Linear Algebra Examples

$\frac{\sqrt{27} a^{3} b c^{2}}{\sqrt{12} a c^{2}}$

Step 1

Cancel the common factor of $a^{3}$ and $a$ .

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Step 1.1

Factor $a$ out of $\sqrt{27} a^{3} b c^{2}$ .

$\frac{a (\sqrt{27} a^{2} b c^{2})}{\sqrt{12} a c^{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $a$ out of $\sqrt{12} a c^{2}$ .

$\frac{a (\sqrt{27} a^{2} b c^{2})}{a (\sqrt{12} c^{2})}$

Step 1.2.2

Cancel the common factor.

$\frac{a (\sqrt{27} a^{2} b c^{2})}{a (\sqrt{12} c^{2})}$

Step 1.2.3

Rewrite the expression.

$\frac{\sqrt{27} a^{2} b c^{2}}{\sqrt{12} c^{2}}$

Step 2

Cancel the common factor of $c^{2}$ .

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Step 2.1

Cancel the common factor.

$\frac{\sqrt{27} a^{2} b c^{2}}{\sqrt{12} c^{2}}$

Step 2.2

Rewrite the expression.

$\frac{\sqrt{27} a^{2} b}{\sqrt{12}}$

Step 3

Combine $\sqrt{27}$ and $\sqrt{12}$ into a single radical.

$\sqrt{\frac{27}{12}} a^{2} b$

Step 4

Cancel the common factor of $27$ and $12$ .

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Step 4.1

Factor $3$ out of $27$ .

$\sqrt{\frac{3 (9)}{12}} a^{2} b$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $12$ .

$\sqrt{\frac{3 \cdot 9}{3 \cdot 4}} a^{2} b$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 9}{3 \cdot 4}} a^{2} b$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{9}{4}} a^{2} b$

Step 5

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$\frac{\sqrt{9}}{\sqrt{4}} a^{2} b$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{\sqrt{3^{2}}}{\sqrt{4}} a^{2} b$

Step 6.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{\sqrt{4}} a^{2} b$

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{2} a^{2} b$

Step 8

Combine fractions.

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Step 8.1

Combine $\frac{3}{2}$ and $a^{2}$ .

$\frac{3 a^{2}}{2} b$

Step 8.2

Combine $\frac{3 a^{2}}{2}$ and $b$ .

$\frac{3 a^{2} b}{2}$