Linear Algebra Examples

$\frac{\sqrt{5} - \sqrt[4]{24}}{\sqrt{3} - \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{5} - \sqrt[4]{24}}{\sqrt{3} - \sqrt{2}}$ by $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}$ .

$\frac{\sqrt{5} - \sqrt[4]{24}}{\sqrt{3} - \sqrt{2}} \cdot \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{5} - \sqrt[4]{24}}{\sqrt{3} - \sqrt{2}}$ by $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}$ .

$\frac{(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2}) (\sqrt{3} + \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})}{{\sqrt{3}}^{2} + \sqrt{6} - \sqrt{6} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})}{1}$

Step 2.4

Divide $(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})$ by $1$ .

$(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})$

Step 3

Expand $(\sqrt{5} - \sqrt[4]{24}) (\sqrt{3} + \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{5} (\sqrt{3} + \sqrt{2}) - \sqrt[4]{24} (\sqrt{3} + \sqrt{2})$

Step 3.2

Apply the distributive property.

$\sqrt{5} \sqrt{3} + \sqrt{5} \sqrt{2} - \sqrt[4]{24} (\sqrt{3} + \sqrt{2})$

Step 3.3

Apply the distributive property.

$\sqrt{5} \sqrt{3} + \sqrt{5} \sqrt{2} - \sqrt[4]{24} \sqrt{3} - \sqrt[4]{24} \sqrt{2}$

Step 4

Simplify each term.

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Step 4.1

Combine using the product rule for radicals.

$\sqrt{5 \cdot 3} + \sqrt{5} \sqrt{2} - \sqrt[4]{24} \sqrt{3} - \sqrt[4]{24} \sqrt{2}$

Step 4.2

Multiply $5$ by $3$ .

$\sqrt{15} + \sqrt{5} \sqrt{2} - \sqrt[4]{24} \sqrt{3} - \sqrt[4]{24} \sqrt{2}$

Step 4.3

Combine using the product rule for radicals.

$\sqrt{15} + \sqrt{5 \cdot 2} - \sqrt[4]{24} \sqrt{3} - \sqrt[4]{24} \sqrt{2}$

Step 4.4

Multiply $5$ by $2$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{24} \sqrt{3} - \sqrt[4]{24} \sqrt{2}$

Step 4.5

Multiply $- \sqrt[4]{24} \sqrt{3}$ .

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Step 4.5.1

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

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Step 4.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{15} + \sqrt{10} - (3^{\frac{1}{2}} \sqrt[4]{24}) - \sqrt[4]{24} \sqrt{2}$

Step 4.5.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$\sqrt{15} + \sqrt{10} - (3^{\frac{2}{4}} \sqrt[4]{24}) - \sqrt[4]{24} \sqrt{2}$

Step 4.5.1.3

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$\sqrt{15} + \sqrt{10} - (\sqrt[4]{3^{2}} \sqrt[4]{24}) - \sqrt[4]{24} \sqrt{2}$

Step 4.5.2

Combine using the product rule for radicals.

$\sqrt{15} + \sqrt{10} - \sqrt[4]{3^{2} \cdot 24} - \sqrt[4]{24} \sqrt{2}$

Step 4.5.3

Raise $3$ to the power of $2$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{9 \cdot 24} - \sqrt[4]{24} \sqrt{2}$

Step 4.5.4

Multiply $9$ by $24$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{24} \sqrt{2}$

Step 4.6

Multiply $- \sqrt[4]{24} \sqrt{2}$ .

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Step 4.6.1

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

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Step 4.6.1.1

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

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - (2^{\frac{1}{2}} \sqrt[4]{24})$

Step 4.6.1.2

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

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - (2^{\frac{2}{4}} \sqrt[4]{24})$

Step 4.6.1.3

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

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - (\sqrt[4]{2^{2}} \sqrt[4]{24})$

Step 4.6.2

Combine using the product rule for radicals.

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{2^{2} \cdot 24}$

Step 4.6.3

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

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{4 \cdot 24}$

Step 4.6.4

Multiply $4$ by $24$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{96}$

Step 4.7

Rewrite $96$ as $2^{4} \cdot 6$ .

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Step 4.7.1

Factor $16$ out of $96$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{16 (6)}$

Step 4.7.2

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

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - \sqrt[4]{2^{4} \cdot 6}$

Step 4.8

Pull terms out from under the radical.

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - (2 \sqrt[4]{6})$

Step 4.9

Multiply $2$ by $- 1$ .

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - 2 \sqrt[4]{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15} + \sqrt{10} - \sqrt[4]{216} - 2 \sqrt[4]{6}$

Decimal Form:

$0.07143322 \dots$