Simplify ( fifth root of square root of 40- square root of 8)( fifth root of square root of 40+ square root of 8)

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

Step 1

Combine using the product rule for radicals.

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

Step 2

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

Factor $4$ out of $40$ .

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

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 $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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 $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

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

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 $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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.

Apply the distributive property.

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

Apply the distributive property.

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

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.

Simplify each term.

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $10$ .

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

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

Multiply $2$ by $2$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $10$ .

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

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

Factor $4$ out of $20$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $4$ .

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

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

Multiply $2$ by $- 2$ .

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

Combine using the product rule for radicals.

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

Multiply $10$ by $2$ .

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

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

Factor $4$ out of $20$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 4$ .

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

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

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 4$ by $2$ .

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

Subtract $8$ from $40$ .

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

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

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

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.

$2$

Simplify ( fifth root of square root of 40- square root of 8)( fifth root of square root of 40+ square root of 8)

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