Algebra Examples

$(\sqrt[5]{\sqrt{40} - \sqrt{8}}) (\sqrt[5]{\sqrt{40} + \sqrt{8}})$

Step 1

Combine using the product rule for radicals.

$\sqrt[5]{(\sqrt{40} - \sqrt{8}) (\sqrt{40} + \sqrt{8})}$

Step 2

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 2.1

Factor $4$ out of $40$ .

$\sqrt[5]{(\sqrt{4 (10)} - \sqrt{8}) (\sqrt{40} + \sqrt{8})}$

Step 2.2

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

$\sqrt[5]{(\sqrt{2^{2} \cdot 10} - \sqrt{8}) (\sqrt{40} + \sqrt{8})}$

Step 3

Pull terms out from under the radical.

$\sqrt[5]{(2 \sqrt{10} - \sqrt{8}) (\sqrt{40} + \sqrt{8})}$

Step 4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1

Factor $4$ out of $8$ .

$\sqrt[5]{(2 \sqrt{10} - \sqrt{4 (2)}) (\sqrt{40} + \sqrt{8})}$

Step 4.2

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

$\sqrt[5]{(2 \sqrt{10} - \sqrt{2^{2} \cdot 2}) (\sqrt{40} + \sqrt{8})}$

Step 5

Pull terms out from under the radical.

$\sqrt[5]{(2 \sqrt{10} - (2 \sqrt{2})) (\sqrt{40} + \sqrt{8})}$

Step 6

Multiply $2$ by $- 1$ .

$\sqrt[5]{(2 \sqrt{10} - 2 \sqrt{2}) (\sqrt{40} + \sqrt{8})}$

Step 7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 7.1

Factor $4$ out of $40$ .

$\sqrt[5]{(2 \sqrt{10} - 2 \sqrt{2}) (\sqrt{4 (10)} + \sqrt{8})}$

Step 7.2

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

$\sqrt[5]{(2 \sqrt{10} - 2 \sqrt{2}) (\sqrt{2^{2} \cdot 10} + \sqrt{8})}$

Step 8

Pull terms out from under the radical.

$\sqrt[5]{(2 \sqrt{10} - 2 \sqrt{2}) (2 \sqrt{10} + \sqrt{8})}$

Step 9

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.1

Factor $4$ out of $8$ .

$\sqrt[5]{(2 \sqrt{10} - 2 \sqrt{2}) (2 \sqrt{10} + \sqrt{4 (2)})}$

Step 9.2

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

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

Step 10

Pull terms out from under the radical.

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

Step 11

Expand $(2 \sqrt{10} - 2 \sqrt{2}) (2 \sqrt{10} + 2 \sqrt{2})$ using the FOIL Method.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Apply the distributive property.

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

Step 12

Simplify and combine like terms.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $2 \sqrt{10} (2 \sqrt{10})$ .

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Step 12.1.1.1

Multiply $2$ by $2$ .

$\sqrt[5]{4 \sqrt{10} \sqrt{10} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.1.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt[5]{4 ({\sqrt{10}}^{1} \sqrt{10}) + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.1.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt[5]{4 ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.1.4

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

$\sqrt[5]{4 {\sqrt{10}}^{1 + 1} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.1.5

Add $1$ and $1$ .

$\sqrt[5]{4 {\sqrt{10}}^{2} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 12.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt[5]{4 {(10^{\frac{1}{2}})}^{2} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[5]{4 \cdot 10^{\frac{1}{2} \cdot 2} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt[5]{4 \cdot 10^{\frac{2}{2}} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2.4

Cancel the common factor of $2$ .

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Step 12.1.2.4.1

Cancel the common factor.

$\sqrt[5]{4 \cdot 10^{\frac{2}{2}} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2.4.2

Rewrite the expression.

$\sqrt[5]{4 \cdot 10^{1} + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.2.5

Evaluate the exponent.

$\sqrt[5]{4 \cdot 10 + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.3

Multiply $4$ by $10$ .

$\sqrt[5]{40 + 2 \sqrt{10} (2 \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.4

Multiply $2 \sqrt{10} (2 \sqrt{2})$ .

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Step 12.1.4.1

Multiply $2$ by $2$ .

$\sqrt[5]{40 + 4 \sqrt{10} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.4.2

Combine using the product rule for radicals.

$\sqrt[5]{40 + 4 \sqrt{2 \cdot 10} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.4.3

Multiply $2$ by $10$ .

$\sqrt[5]{40 + 4 \sqrt{20} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.5

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 12.1.5.1

Factor $4$ out of $20$ .

$\sqrt[5]{40 + 4 \sqrt{4 (5)} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.5.2

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

$\sqrt[5]{40 + 4 \sqrt{2^{2} \cdot 5} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.6

Pull terms out from under the radical.

$\sqrt[5]{40 + 4 (2 \sqrt{5}) - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.7

Multiply $2$ by $4$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 2 \sqrt{2} (2 \sqrt{10}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.8

Multiply $- 2 \sqrt{2} (2 \sqrt{10})$ .

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Step 12.1.8.1

Multiply $2$ by $- 2$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 4 \sqrt{2} \sqrt{10} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.8.2

Combine using the product rule for radicals.

$\sqrt[5]{40 + 8 \sqrt{5} - 4 \sqrt{10 \cdot 2} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.8.3

Multiply $10$ by $2$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 4 \sqrt{20} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.9

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 12.1.9.1

Factor $4$ out of $20$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 4 \sqrt{4 (5)} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.9.2

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

$\sqrt[5]{40 + 8 \sqrt{5} - 4 \sqrt{2^{2} \cdot 5} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.10

Pull terms out from under the radical.

$\sqrt[5]{40 + 8 \sqrt{5} - 4 (2 \sqrt{5}) - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.11

Multiply $2$ by $- 4$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 2 \sqrt{2} (2 \sqrt{2})}$

Step 12.1.12

Multiply $- 2 \sqrt{2} (2 \sqrt{2})$ .

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Step 12.1.12.1

Multiply $2$ by $- 2$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \sqrt{2} \sqrt{2}}$

Step 12.1.12.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 12.1.12.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 12.1.12.4

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

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 {\sqrt{2}}^{1 + 1}}$

Step 12.1.12.5

Add $1$ and $1$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 {\sqrt{2}}^{2}}$

Step 12.1.13

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 12.1.13.1

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

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 {(2^{\frac{1}{2}})}^{2}}$

Step 12.1.13.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 12.1.13.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \cdot 2^{\frac{2}{2}}}$

Step 12.1.13.4

Cancel the common factor of $2$ .

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Step 12.1.13.4.1

Cancel the common factor.

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \cdot 2^{\frac{2}{2}}}$

Step 12.1.13.4.2

Rewrite the expression.

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \cdot 2^{1}}$

Step 12.1.13.5

Evaluate the exponent.

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 4 \cdot 2}$

Step 12.1.14

Multiply $- 4$ by $2$ .

$\sqrt[5]{40 + 8 \sqrt{5} - 8 \sqrt{5} - 8}$

Step 12.2

Subtract $8$ from $40$ .

$\sqrt[5]{32 + 8 \sqrt{5} - 8 \sqrt{5}}$

Step 12.3

Subtract $8 \sqrt{5}$ from $8 \sqrt{5}$ .

$\sqrt[5]{32 + 0}$

Step 12.4

Add $32$ and $0$ .

$\sqrt[5]{32}$

Step 13

Rewrite $32$ as $2^{5}$ .

$\sqrt[5]{2^{5}}$

Step 14

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

$2$