Linear Algebra Examples

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

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

Step 1

Combine

\sqrt{c^{3}}

$\sqrt{c^{3}}$ and

\sqrt{c}

$\sqrt{c}$ into a single radical.

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

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

Step 2

Cancel the common factor of

c^{3}

$c^{3}$ and

c

$c$ .

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Step 2.1

Factor

c

$c$ out of

c^{3}

$c^{3}$ .

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Raise

c

$c$ to the power of

1

$1$ .

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

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

Step 2.2.2

Factor

c

$c$ out of

c^{1}

$c^{1}$ .

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

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

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}}

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

Step 2.2.4

Rewrite the expression.

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

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

Step 2.2.5

Divide

c^{2}

$c^{2}$ by

1

$1$ .

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

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

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

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

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

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

Step 3

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

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

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to

2 a

$2 a$ .

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

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

Step 4.2

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of

8

$8$ and

4

$4$ .

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Step 5.1.1

Factor

4

$4$ out of

8 a^{2} b c

$8 a^{2} b c$ .

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

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

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor

4

$4$ out of

4 a^{2} \sqrt{b}

$4 a^{2} \sqrt{b}$ .

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

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

Step 5.1.2.3

Rewrite the expression.

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

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

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

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

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

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

Step 5.2

Cancel the common factor of

a^{2}

$a^{2}$ .

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Step 5.2.1

Cancel the common factor.

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

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

Step 5.2.2

Rewrite the expression.

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

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

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

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

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

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

Step 6

Multiply

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

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

\frac{\sqrt{b}}{\sqrt{b}}

$\frac{\sqrt{b}}{\sqrt{b}}$ .

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

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

\frac{\sqrt{b}}{\sqrt{b}}

$\frac{\sqrt{b}}{\sqrt{b}}$ .

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

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

Step 7.2

Raise

\sqrt{b}

$\sqrt{b}$ to the power of

1

$1$ .

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

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

Step 7.3

Raise

\sqrt{b}

$\sqrt{b}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 7.5

Add

1

$1$ and

1

$1$ .

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

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

Step 7.6

Rewrite

{\sqrt{b}}^{2}

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

b

$b$ .

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Step 7.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{b}

$\sqrt{b}$ as

b^{\frac{1}{2}}

$b^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 7.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 7.6.4

Cancel the common factor of

2

$2$ .

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Step 7.6.4.1

Cancel the common factor.

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

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

Step 7.6.4.2

Rewrite the expression.

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

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

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

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

Step 7.6.5

Simplify.

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

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

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

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

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

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

Step 8

Cancel the common factor of

b

$b$ .

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Step 8.1

Cancel the common factor.

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

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

Step 8.2

Divide

2 c \sqrt{b}

$2 c \sqrt{b}$ by

1

$1$ .

2 c \sqrt{b}

$2 c \sqrt{b}$

2 c \sqrt{b}

$2 c \sqrt{b}$