Linear Algebra Examples

$\frac{8 a^{2} b \sqrt{c^{3}}}{{(2 a)}^{2} \sqrt{b} \sqrt{c}}$

Step 1

Combine $\sqrt{c^{3}}$ and $\sqrt{c}$ into a single radical.

$\frac{8 a^{2} b \sqrt{\frac{c^{3}}{c}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2

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

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Step 2.1

Factor $c$ out of $c^{3}$ .

$\frac{8 a^{2} b \sqrt{\frac{c \cdot c^{2}}{c}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$\frac{8 a^{2} b \sqrt{\frac{c \cdot c^{2}}{c^{1}}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2.2.2

Factor $c$ out of $c^{1}$ .

$\frac{8 a^{2} b \sqrt{\frac{c \cdot c^{2}}{c \cdot 1}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2.2.3

Cancel the common factor.

$\frac{8 a^{2} b \sqrt{\frac{c \cdot c^{2}}{c \cdot 1}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2.2.4

Rewrite the expression.

$\frac{8 a^{2} b \sqrt{\frac{c^{2}}{1}}}{{(2 a)}^{2} \sqrt{b}}$

Step 2.2.5

Divide $c^{2}$ by $1$ .

$\frac{8 a^{2} b \sqrt{c^{2}}}{{(2 a)}^{2} \sqrt{b}}$

Step 3

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

$\frac{8 a^{2} b c}{{(2 a)}^{2} \sqrt{b}}$

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $2 a$ .

$\frac{8 a^{2} b c}{2^{2} a^{2} \sqrt{b}}$

Step 4.2

Raise $2$ to the power of $2$ .

$\frac{8 a^{2} b c}{4 a^{2} \sqrt{b}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $8$ and $4$ .

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Step 5.1.1

Factor $4$ out of $8 a^{2} b c$ .

$\frac{4 (2 a^{2} b c)}{4 a^{2} \sqrt{b}}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $4$ out of $4 a^{2} \sqrt{b}$ .

$\frac{4 (2 a^{2} b c)}{4 (a^{2} \sqrt{b})}$

Step 5.1.2.2

Cancel the common factor.

$\frac{4 (2 a^{2} b c)}{4 (a^{2} \sqrt{b})}$

Step 5.1.2.3

Rewrite the expression.

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

Step 5.2

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

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

$\frac{2 b c}{\sqrt{b}}$

Step 6

Multiply $\frac{2 b c}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{2 b c}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2 b c}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{2 b c \sqrt{b}}{\sqrt{b} \sqrt{b}}$

Step 7.2

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

$\frac{2 b c \sqrt{b}}{{\sqrt{b}}^{1} \sqrt{b}}$

Step 7.3

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

$\frac{2 b c \sqrt{b}}{{\sqrt{b}}^{1} {\sqrt{b}}^{1}}$

Step 7.4

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

$\frac{2 b c \sqrt{b}}{{\sqrt{b}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{2 b c \sqrt{b}}{{\sqrt{b}}^{2}}$

Step 7.6

Rewrite ${\sqrt{b}}^{2}$ as $b$ .

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Step 7.6.1

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

$\frac{2 b c \sqrt{b}}{{(b^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{2 b c \sqrt{b}}{b^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{2 b c \sqrt{b}}{b^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{2 b c \sqrt{b}}{b^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{2 b c \sqrt{b}}{b^{1}}$

Step 7.6.5

Simplify.

$\frac{2 b c \sqrt{b}}{b}$

Step 8

Cancel the common factor of $b$ .

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Step 8.1

Cancel the common factor.

$\frac{2 b c \sqrt{b}}{b}$

Step 8.2

Divide $2 c \sqrt{b}$ by $1$ .

$2 c \sqrt{b}$