Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Multiply $\sqrt[3]{5} \sqrt{5}$ .

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Step 2.1.1

Rewrite the expression using the least common index of $6$ .

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Step 2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$5^{\frac{1}{3}} \sqrt{5} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.1.2

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{2}{6}}$ .

$5^{\frac{2}{6}} \sqrt{5} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.1.3

Rewrite $5^{\frac{2}{6}}$ as $\sqrt[6]{5^{2}}$ .

$\sqrt[6]{5^{2}} \sqrt{5} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.1.4

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

$\sqrt[6]{5^{2}} \cdot 5^{\frac{1}{2}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.1.5

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

$\sqrt[6]{5^{2}} \cdot 5^{\frac{3}{6}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.1.6

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

$\sqrt[6]{5^{2}} \sqrt[6]{5^{3}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.2

Combine using the product rule for radicals.

$\sqrt[6]{5^{2} \cdot 5^{3}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.3

Multiply $5^{2}$ by $5^{3}$ by adding the exponents.

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Step 2.1.3.1

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

$\sqrt[6]{5^{2 + 3}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.1.3.2

Add $2$ and $3$ .

$\sqrt[6]{5^{5}} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.2

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

$\sqrt[6]{3125} + \sqrt[3]{5} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3

Multiply $\sqrt[3]{5} \sqrt{2}$ .

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Step 2.3.1

Rewrite the expression using the least common index of $6$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\sqrt[6]{3125} + 5^{\frac{1}{3}} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.1.2

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{2}{6}}$ .

$\sqrt[6]{3125} + 5^{\frac{2}{6}} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.1.3

Rewrite $5^{\frac{2}{6}}$ as $\sqrt[6]{5^{2}}$ .

$\sqrt[6]{3125} + \sqrt[6]{5^{2}} \sqrt{2} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.1.4

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

$\sqrt[6]{3125} + \sqrt[6]{5^{2}} \cdot 2^{\frac{1}{2}} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.1.5

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

$\sqrt[6]{3125} + \sqrt[6]{5^{2}} \cdot 2^{\frac{3}{6}} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.1.6

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$\sqrt[6]{3125} + \sqrt[6]{5^{2}} \sqrt[6]{2^{3}} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.2

Combine using the product rule for radicals.

$\sqrt[6]{3125} + \sqrt[6]{5^{2} \cdot 2^{3}} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.3

Raise $5$ to the power of $2$ .

$\sqrt[6]{3125} + \sqrt[6]{25 \cdot 2^{3}} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.4

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

$\sqrt[6]{3125} + \sqrt[6]{25 \cdot 8} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.3.5

Multiply $25$ by $8$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2} \sqrt{5} + \sqrt[4]{2} \sqrt{2}$

Step 2.4

Multiply $\sqrt[4]{2} \sqrt{5}$ .

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Step 2.4.1

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

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Step 2.4.1.1

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2} \cdot 5^{\frac{1}{2}} + \sqrt[4]{2} \sqrt{2}$

Step 2.4.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2} \cdot 5^{\frac{2}{4}} + \sqrt[4]{2} \sqrt{2}$

Step 2.4.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2} \sqrt[4]{5^{2}} + \sqrt[4]{2} \sqrt{2}$

Step 2.4.2

Combine using the product rule for radicals.

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2 \cdot 5^{2}} + \sqrt[4]{2} \sqrt{2}$

Step 2.4.3

Raise $5$ to the power of $2$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{2 \cdot 25} + \sqrt[4]{2} \sqrt{2}$

Step 2.4.4

Multiply $2$ by $25$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2} \sqrt{2}$

Step 2.5

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

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Step 2.5.1

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

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Step 2.5.1.1

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2} \cdot 2^{\frac{1}{2}}$

Step 2.5.1.2

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2} \cdot 2^{\frac{2}{4}}$

Step 2.5.1.3

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2} \sqrt[4]{2^{2}}$

Step 2.5.2

Combine using the product rule for radicals.

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2 \cdot 2^{2}}$

Step 2.5.3

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

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Step 2.5.3.1

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

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Step 2.5.3.1.1

Raise $2$ to the power of $1$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2^{1} \cdot 2^{2}}$

Step 2.5.3.1.2

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2^{1 + 2}}$

Step 2.5.3.2

Add $1$ and $2$ .

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{2^{3}}$

Step 2.6

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

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{8}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{3125} + \sqrt[6]{200} + \sqrt[4]{50} + \sqrt[4]{8}$

Decimal Form:

$10.58283441 \dots$