Simplify ( cube root of square root of 30+ square root of 3)( cube root of square root of 30- square root of 3)

$(\sqrt[3]{\sqrt{30} + \sqrt{3}}) (\sqrt[3]{\sqrt{30} - \sqrt{3}})$

Step 1

Combine using the product rule for radicals.

$\sqrt[3]{(\sqrt{30} + \sqrt{3}) (\sqrt{30} - \sqrt{3})}$

Step 2

Expand $(\sqrt{30} + \sqrt{3}) (\sqrt{30} - \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$\sqrt[3]{\sqrt{30} (\sqrt{30} - \sqrt{3}) + \sqrt{3} (\sqrt{30} - \sqrt{3})}$

Apply the distributive property.

$\sqrt[3]{\sqrt{30} \sqrt{30} + \sqrt{30} (- \sqrt{3}) + \sqrt{3} (\sqrt{30} - \sqrt{3})}$

Apply the distributive property.

$\sqrt[3]{\sqrt{30} \sqrt{30} + \sqrt{30} (- \sqrt{3}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt[3]{\sqrt{30 \cdot 30} + \sqrt{30} (- \sqrt{3}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Multiply $30$ by $30$ .

$\sqrt[3]{\sqrt{900} + \sqrt{30} (- \sqrt{3}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Rewrite $900$ as $30^{2}$ .

$\sqrt[3]{\sqrt{30^{2}} + \sqrt{30} (- \sqrt{3}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

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

$\sqrt[3]{30 + \sqrt{30} (- \sqrt{3}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Multiply $\sqrt{30} (- \sqrt{3})$ .

Combine using the product rule for radicals.

$\sqrt[3]{30 - \sqrt{30 \cdot 3} + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Multiply $30$ by $3$ .

$\sqrt[3]{30 - \sqrt{90} + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$\sqrt[3]{30 - \sqrt{9 (10)} + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Rewrite $9$ as $3^{2}$ .

$\sqrt[3]{30 - \sqrt{3^{2} \cdot 10} + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Pull terms out from under the radical.

$\sqrt[3]{30 - (3 \sqrt{10}) + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Multiply $3$ by $- 1$ .

$\sqrt[3]{30 - 3 \sqrt{10} + \sqrt{3} \sqrt{30} + \sqrt{3} (- \sqrt{3})}$

Combine using the product rule for radicals.

$\sqrt[3]{30 - 3 \sqrt{10} + \sqrt{3 \cdot 30} + \sqrt{3} (- \sqrt{3})}$

Multiply $3$ by $30$ .

$\sqrt[3]{30 - 3 \sqrt{10} + \sqrt{90} + \sqrt{3} (- \sqrt{3})}$

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$\sqrt[3]{30 - 3 \sqrt{10} + \sqrt{9 (10)} + \sqrt{3} (- \sqrt{3})}$

Rewrite $9$ as $3^{2}$ .

$\sqrt[3]{30 - 3 \sqrt{10} + \sqrt{3^{2} \cdot 10} + \sqrt{3} (- \sqrt{3})}$

Pull terms out from under the radical.

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} + \sqrt{3} (- \sqrt{3})}$

Multiply $\sqrt{3} (- \sqrt{3})$ .

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - ({\sqrt{3}}^{1} \sqrt{3})}$

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

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

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - {\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 3^{1}}$

Evaluate the exponent.

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 1 \cdot 3}$

Multiply $- 1$ by $3$ .

$\sqrt[3]{30 - 3 \sqrt{10} + 3 \sqrt{10} - 3}$

Subtract $3$ from $30$ .

$\sqrt[3]{27 - 3 \sqrt{10} + 3 \sqrt{10}}$

Add $- 3 \sqrt{10}$ and $3 \sqrt{10}$ .

$\sqrt[3]{27 + 0}$

Add $27$ and $0$ .

$\sqrt[3]{27}$

Step 4

Rewrite $27$ as $3^{3}$ .

$\sqrt[3]{3^{3}}$

Step 5

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

$3$

Simplify ( cube root of square root of 30+ square root of 3)( cube root of square root of 30- square root of 3)

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