Finite Math Examples

$\sqrt[4]{32 b c^{5}} \sqrt{2 b^{3} c}$

Step 1

Rewrite $32 b c^{5}$ as ${(2 c)}^{4} \cdot (2 b c)$ .

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Step 1.1

Factor $16$ out of $32$ .

$\sqrt[4]{16 (2) b c^{5}} \sqrt{2 b^{3} c}$

Step 1.2

Rewrite $16$ as $2^{4}$ .

$\sqrt[4]{2^{4} \cdot 2 b c^{5}} \sqrt{2 b^{3} c}$

Step 1.3

Factor out $c^{4}$ .

$\sqrt[4]{2^{4} \cdot 2 b (c^{4} c)} \sqrt{2 b^{3} c}$

Step 1.4

Move $b$ .

$\sqrt[4]{2^{4} \cdot 2 c^{4} b c} \sqrt{2 b^{3} c}$

Step 1.5

Move $2$ .

$\sqrt[4]{2^{4} c^{4} \cdot 2 b c} \sqrt{2 b^{3} c}$

Step 1.6

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

$\sqrt[4]{{(2 c)}^{4} \cdot 2 b c} \sqrt{2 b^{3} c}$

Step 1.7

Add parentheses.

$\sqrt[4]{{(2 c)}^{4} \cdot 2 (b c)} \sqrt{2 b^{3} c}$

Step 1.8

Add parentheses.

$\sqrt[4]{{(2 c)}^{4} \cdot (2 b c)} \sqrt{2 b^{3} c}$

Step 2

Pull terms out from under the radical.

$2 c \sqrt[4]{2 b c} \sqrt{2 b^{3} c}$

Step 3

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

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Step 3.1

Factor out $b^{2}$ .

$2 c \sqrt[4]{2 b c} \sqrt{2 (b^{2} b) c}$

Step 3.2

Reorder $2$ and $b^{2}$ .

$2 c \sqrt[4]{2 b c} \sqrt{b^{2} \cdot 2 b c}$

Step 3.3

Add parentheses.

$2 c \sqrt[4]{2 b c} \sqrt{b^{2} \cdot 2 (b c)}$

Step 3.4

Add parentheses.

$2 c \sqrt[4]{2 b c} \sqrt{b^{2} \cdot (2 b c)}$

Step 4

Pull terms out from under the radical.

$2 c \sqrt[4]{2 b c} (b \sqrt{2 b c})$

Step 5

Multiply $2 c \sqrt[4]{2 b c} (b \sqrt{2 b c})$ .

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Step 5.1

Rewrite the expression using the least common index of $4$ .

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Step 5.1.1

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

$2 c (b (\sqrt[4]{2 b c} {(2 b c)}^{\frac{1}{2}}))$

Step 5.1.2

Rewrite ${(2 b c)}^{\frac{1}{2}}$ as ${(2 b c)}^{\frac{2}{4}}$ .

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

Step 5.1.3

Rewrite ${(2 b c)}^{\frac{2}{4}}$ as $\sqrt[4]{{(2 b c)}^{2}}$ .

$2 c (b (\sqrt[4]{2 b c} \sqrt[4]{{(2 b c)}^{2}}))$

Step 5.2

Combine using the product rule for radicals.

$2 c (b \sqrt[4]{2 b c {(2 b c)}^{2}})$

Step 5.3

Apply the product rule to $2 b c$ .

$2 c (b \sqrt[4]{2 b c {(2 b)}^{2} c^{2}})$

Step 5.4

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

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Step 5.4.1

Move $c^{2}$ .

$2 c (b \sqrt[4]{2 b (c^{2} c) {(2 b)}^{2}})$

Step 5.4.2

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

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Step 5.4.2.1

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

$2 c (b \sqrt[4]{2 b (c^{2} c^{1}) {(2 b)}^{2}})$

Step 5.4.2.2

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

$2 c (b \sqrt[4]{2 b c^{2 + 1} {(2 b)}^{2}})$

Step 5.4.3

Add $2$ and $1$ .

$2 c (b \sqrt[4]{2 b c^{3} {(2 b)}^{2}})$

Step 5.5

Apply the product rule to $2 b$ .

$2 c (b \sqrt[4]{2 b c^{3} (2^{2} b^{2})})$

Step 5.6

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

$2 c (b \sqrt[4]{2 b c^{3} \cdot 4 b^{2}})$

Step 5.7

Combine exponents.

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Step 5.7.1

Multiply $4$ by $2$ .

$2 c (b \sqrt[4]{8 b c^{3} b^{2}})$

Step 5.7.2

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

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Step 5.7.2.1

Move $b^{2}$ .

$2 c (b \sqrt[4]{8 (b^{2} b) c^{3}})$

Step 5.7.2.2

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

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Step 5.7.2.2.1

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

$2 c (b \sqrt[4]{8 (b^{2} b^{1}) c^{3}})$

Step 5.7.2.2.2

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

$2 c (b \sqrt[4]{8 b^{2 + 1} c^{3}})$

Step 5.7.2.3

Add $2$ and $1$ .

$2 c (b \sqrt[4]{8 b^{3} c^{3}})$

Step 6

Reorder factors in $2 c (b \sqrt[4]{8 b^{3} c^{3}})$ .

$2 c b \sqrt[4]{8 b^{3} c^{3}}$