Finite Math Examples

$(3 \sqrt[4]{3} + \sqrt[4]{7}) (2 \sqrt[4]{4} - 5 \sqrt[4]{6})$

Step 1

Simplify each term.

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Step 1.1

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

$(3 \sqrt[4]{3} + \sqrt[4]{7}) (2 \sqrt[4]{2^{2}} - 5 \sqrt[4]{6})$

Step 1.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$(3 \sqrt[4]{3} + \sqrt[4]{7}) (2 \sqrt{\sqrt{2^{2}}} - 5 \sqrt[4]{6})$

Step 1.3

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

$(3 \sqrt[4]{3} + \sqrt[4]{7}) (2 \sqrt{2} - 5 \sqrt[4]{6})$

Step 2

Expand $(3 \sqrt[4]{3} + \sqrt[4]{7}) (2 \sqrt{2} - 5 \sqrt[4]{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 \sqrt[4]{3} (2 \sqrt{2} - 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2} - 5 \sqrt[4]{6})$

Step 2.2

Apply the distributive property.

$3 \sqrt[4]{3} (2 \sqrt{2}) + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2} - 5 \sqrt[4]{6})$

Step 2.3

Apply the distributive property.

$3 \sqrt[4]{3} (2 \sqrt{2}) + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3

Simplify each term.

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Step 3.1

Multiply $3 \sqrt[4]{3} (2 \sqrt{2})$ .

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Step 3.1.1

Multiply $2$ by $3$ .

$6 \sqrt[4]{3} \sqrt{2} + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.2

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

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Step 3.1.2.1

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

$6 (2^{\frac{1}{2}} \sqrt[4]{3}) + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.2.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$6 (2^{\frac{2}{4}} \sqrt[4]{3}) + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.2.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$6 (\sqrt[4]{2^{2}} \sqrt[4]{3}) + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.3

Combine using the product rule for radicals.

$6 \sqrt[4]{2^{2} \cdot 3} + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.4

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

$6 \sqrt[4]{4 \cdot 3} + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.1.5

Multiply $4$ by $3$ .

$6 \sqrt[4]{12} + 3 \sqrt[4]{3} (- 5 \sqrt[4]{6}) + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.2

Multiply $3 \sqrt[4]{3} (- 5 \sqrt[4]{6})$ .

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Step 3.2.1

Multiply $- 5$ by $3$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{3} \sqrt[4]{6} + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.2.2

Combine using the product rule for radicals.

$6 \sqrt[4]{12} - 15 \sqrt[4]{6 \cdot 3} + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.2.3

Multiply $6$ by $3$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + \sqrt[4]{7} (2 \sqrt{2}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3

Multiply $\sqrt[4]{7} (2 \sqrt{2})$ .

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Step 3.3.1

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

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Step 3.3.1.1

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

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 (\sqrt[4]{7} \cdot 2^{\frac{1}{2}}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 (\sqrt[4]{7} \cdot 2^{\frac{2}{4}}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3.1.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 (\sqrt[4]{7} \sqrt[4]{2^{2}}) + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3.2

Combine using the product rule for radicals.

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{7 \cdot 2^{2}} + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3.3

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

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{7 \cdot 4} + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.3.4

Multiply $7$ by $4$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{28} + \sqrt[4]{7} (- 5 \sqrt[4]{6})$

Step 3.4

Multiply $\sqrt[4]{7} (- 5 \sqrt[4]{6})$ .

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Step 3.4.1

Combine using the product rule for radicals.

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{28} - 5 \sqrt[4]{7 \cdot 6}$

Step 3.4.2

Multiply $7$ by $6$ .

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{28} - 5 \sqrt[4]{42}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt[4]{12} - 15 \sqrt[4]{18} + 2 \sqrt[4]{28} - 5 \sqrt[4]{42}$

Decimal Form:

$- 27.85724505 \dots$