Basic Math Examples

$(\sqrt[5]{\sqrt{6} - \sqrt{5}}) (\sqrt[5]{\sqrt{6}} + \sqrt{5})$

Step 1

Rewrite $\sqrt[5]{\sqrt{6}}$ as $\sqrt[10]{6}$ .

$\sqrt[5]{\sqrt{6} - \sqrt{5}} (\sqrt[10]{6} + \sqrt{5})$

Step 2

Apply the distributive property.

$\sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt[10]{6} + \sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$

Step 3

Multiply $\sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt[10]{6}$ .

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Step 3.1

Rewrite the expression using the least common index of $10$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\sqrt{6} - \sqrt{5}}$ as ${(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}}$ .

${(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}} \sqrt[10]{6} + \sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$

Step 3.1.2

Rewrite ${(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}}$ as ${(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}}$ .

${(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}} \sqrt[10]{6} + \sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$

Step 3.1.3

Rewrite ${(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}}$ as $\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}} \sqrt[10]{6} + \sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$

Step 4

Multiply $\sqrt[5]{\sqrt{6} - \sqrt{5}} \sqrt{5}$ .

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Step 4.1

Rewrite the expression using the least common index of $10$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\sqrt{6} - \sqrt{5}}$ as ${(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + {(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}} \sqrt{5}$

Step 4.1.2

Rewrite ${(\sqrt{6} - \sqrt{5})}^{\frac{1}{5}}$ as ${(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + {(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}} \sqrt{5}$

Step 4.1.3

Rewrite ${(\sqrt{6} - \sqrt{5})}^{\frac{2}{10}}$ as $\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}} \sqrt{5}$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}} \cdot 5^{\frac{1}{2}}$

Step 4.1.5

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{5}{10}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}} \cdot 5^{\frac{5}{10}}$

Step 4.1.6

Rewrite $5^{\frac{5}{10}}$ as $\sqrt[10]{5^{5}}$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2}} \sqrt[10]{5^{5}}$

Step 4.2

Combine using the product rule for radicals.

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 5^{5}}$

Step 4.3

Raise $5$ to the power of $5$ .

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 3125}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 6} + \sqrt[10]{{(\sqrt{6} - \sqrt{5})}^{2} \cdot 3125}$

Decimal Form:

$2.52018764 \dots$