Simplify (( square root of 6+ square root of 15+ square root of 3)/( fourth root of 4+ square root of 5+5^0))^2

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt[4]{4} + \sqrt{5} + 5^{0}})}^{2}$

Step 1

Simplify the denominator.

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

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt[4]{2^{2}} + \sqrt{5} + 5^{0}})}^{2}$

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

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{\sqrt{2^{2}}} + \sqrt{5} + 5^{0}})}^{2}$

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

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{2} + \sqrt{5} + 5^{0}})}^{2}$

Anything raised to $0$ is $1$ .

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{2} + \sqrt{5} + 1})}^{2}$

Step 2

Multiply $\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{2} + \sqrt{5} + 1}$ by $\frac{\sqrt{2} + \sqrt{5} - 1}{\sqrt{2} + \sqrt{5} - 1}$ .

${(\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{2} + \sqrt{5} + 1} \cdot \frac{\sqrt{2} + \sqrt{5} - 1}{\sqrt{2} + \sqrt{5} - 1})}^{2}$

Step 3

Multiply $\frac{\sqrt{6} + \sqrt{15} + \sqrt{3}}{\sqrt{2} + \sqrt{5} + 1}$ by $\frac{\sqrt{2} + \sqrt{5} - 1}{\sqrt{2} + \sqrt{5} - 1}$ .

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{(\sqrt{2} + \sqrt{5} + 1) (\sqrt{2} + \sqrt{5} - 1)})}^{2}$

Step 4

Expand the denominator using the FOIL method.

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{{\sqrt{2}}^{2} + \sqrt{10} + \sqrt{2} \cdot - 1 + \sqrt{10} + {\sqrt{5}}^{2} + \sqrt{5} \cdot - 1 + \sqrt{2} + \sqrt{5} - 1})}^{2}$

Step 5

Simplify.

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{6 + 2 \sqrt{10}})}^{2}$

Step 6

Multiply $\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{6 + 2 \sqrt{10}}$ by $\frac{6 - 2 \sqrt{10}}{6 - 2 \sqrt{10}}$ .

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{6 + 2 \sqrt{10}} \cdot \frac{6 - 2 \sqrt{10}}{6 - 2 \sqrt{10}})}^{2}$

Step 7

Simplify terms.

Multiply $\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)}{6 + 2 \sqrt{10}}$ by $\frac{6 - 2 \sqrt{10}}{6 - 2 \sqrt{10}}$ .

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (6 - 2 \sqrt{10})}{(6 + 2 \sqrt{10}) (6 - 2 \sqrt{10})})}^{2}$

Expand the denominator using the FOIL method.

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (6 - 2 \sqrt{10})}{36 - 12 \sqrt{10} + 12 \sqrt{10} - 4 {\sqrt{10}}^{2}})}^{2}$

Simplify.

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (6 - 2 \sqrt{10})}{- 4})}^{2}$

Cancel the common factor of $6 - 2 \sqrt{10}$ and $- 4$ .

Factor $2$ out of $(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (6 - 2 \sqrt{10})$ .

${(\frac{2 ((\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (3 - \sqrt{10}))}{- 4})}^{2}$

Cancel the common factors.

Factor $2$ out of $- 4$ .

${(\frac{2 ((\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (3 - \sqrt{10}))}{2 (- 2)})}^{2}$

Cancel the common factor.

${(\frac{2 ((\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (3 - \sqrt{10}))}{2 \cdot - 2})}^{2}$

Rewrite the expression.

${(\frac{(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Step 8

Simplify the numerator.

Expand $(\sqrt{6} + \sqrt{15} + \sqrt{3}) (\sqrt{2} + \sqrt{5} - 1)$ by multiplying each term in the first expression by each term in the second expression.

${(\frac{(\sqrt{6} \sqrt{2} + \sqrt{6} \sqrt{5} + \sqrt{6} \cdot - 1 + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{6 \cdot 2} + \sqrt{6} \sqrt{5} + \sqrt{6} \cdot - 1 + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $6$ by $2$ .

${(\frac{(\sqrt{12} + \sqrt{6} \sqrt{5} + \sqrt{6} \cdot - 1 + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{12} + \sqrt{6 \cdot 5} + \sqrt{6} \cdot - 1 + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $6$ by $5$ .

${(\frac{(\sqrt{12} + \sqrt{30} + \sqrt{6} \cdot - 1 + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Move $- 1$ to the left of $\sqrt{6}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - 1 \cdot \sqrt{6} + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Rewrite $- 1 \sqrt{6}$ as $- \sqrt{6}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{15} \sqrt{2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{15 \cdot 2} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $15$ by $2$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{15} \sqrt{5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{15 \cdot 5} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $15$ by $5$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} + \sqrt{15} \cdot - 1 + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Move $- 1$ to the left of $\sqrt{15}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - 1 \cdot \sqrt{15} + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Rewrite $- 1 \sqrt{15}$ as $- \sqrt{15}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{3 \cdot 2} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $3$ by $2$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{3} \sqrt{5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{3 \cdot 5} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Multiply $3$ by $5$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} + \sqrt{3} \cdot - 1) (3 - \sqrt{10})}{- 2})}^{2}$

Move $- 1$ to the left of $\sqrt{3}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} - 1 \cdot \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

${(\frac{(\sqrt{12} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Simplify each term.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

${(\frac{(\sqrt{4 (3)} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

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

${(\frac{(\sqrt{2^{2} \cdot 3} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Pull terms out from under the radical.

${(\frac{(2 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{75} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

${(\frac{(2 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{25 (3)} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Rewrite $25$ as $5^{2}$ .

${(\frac{(2 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} + \sqrt{5^{2} \cdot 3} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Pull terms out from under the radical.

${(\frac{(2 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} + 5 \sqrt{3} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Add $2 \sqrt{3}$ and $5 \sqrt{3}$ .

${(\frac{(7 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} - \sqrt{15} + \sqrt{6} + \sqrt{15} - \sqrt{3}) (3 - \sqrt{10})}{- 2})}^{2}$

Subtract $\sqrt{3}$ from $7 \sqrt{3}$ .

${(\frac{(6 \sqrt{3} + \sqrt{30} - \sqrt{6} + \sqrt{30} - \sqrt{15} + \sqrt{6} + \sqrt{15}) (3 - \sqrt{10})}{- 2})}^{2}$

Add $\sqrt{30}$ and $\sqrt{30}$ .

${(\frac{(6 \sqrt{3} + 2 \sqrt{30} - \sqrt{6} - \sqrt{15} + \sqrt{6} + \sqrt{15}) (3 - \sqrt{10})}{- 2})}^{2}$

Add $- \sqrt{6}$ and $\sqrt{6}$ .

${(\frac{(6 \sqrt{3} + 2 \sqrt{30} + 0 - \sqrt{15} + \sqrt{15}) (3 - \sqrt{10})}{- 2})}^{2}$

Add $6 \sqrt{3}$ and $0$ .

${(\frac{(6 \sqrt{3} + 2 \sqrt{30} - \sqrt{15} + \sqrt{15}) (3 - \sqrt{10})}{- 2})}^{2}$

Add $- \sqrt{15}$ and $\sqrt{15}$ .

${(\frac{(6 \sqrt{3} + 2 \sqrt{30} + 0) (3 - \sqrt{10})}{- 2})}^{2}$

Add $6 \sqrt{3}$ and $0$ .

${(\frac{(6 \sqrt{3} + 2 \sqrt{30}) (3 - \sqrt{10})}{- 2})}^{2}$

Expand $(6 \sqrt{3} + 2 \sqrt{30}) (3 - \sqrt{10})$ using the FOIL Method.

Apply the distributive property.

${(\frac{6 \sqrt{3} (3 - \sqrt{10}) + 2 \sqrt{30} (3 - \sqrt{10})}{- 2})}^{2}$

Apply the distributive property.

${(\frac{6 \sqrt{3} \cdot 3 + 6 \sqrt{3} (- \sqrt{10}) + 2 \sqrt{30} (3 - \sqrt{10})}{- 2})}^{2}$

Apply the distributive property.

${(\frac{6 \sqrt{3} \cdot 3 + 6 \sqrt{3} (- \sqrt{10}) + 2 \sqrt{30} \cdot 3 + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $6$ .

${(\frac{18 \sqrt{3} + 6 \sqrt{3} (- \sqrt{10}) + 2 \sqrt{30} \cdot 3 + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Multiply $6 \sqrt{3} (- \sqrt{10})$ .

Multiply $- 1$ by $6$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{3} \sqrt{10} + 2 \sqrt{30} \cdot 3 + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{18 \sqrt{3} - 6 \sqrt{10 \cdot 3} + 2 \sqrt{30} \cdot 3 + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Multiply $10$ by $3$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 2 \sqrt{30} \cdot 3 + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Multiply $3$ by $2$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} + 2 \sqrt{30} (- \sqrt{10})}{- 2})}^{2}$

Multiply $2 \sqrt{30} (- \sqrt{10})$ .

Multiply $- 1$ by $2$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 \sqrt{30} \sqrt{10}}{- 2})}^{2}$

Combine using the product rule for radicals.

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 \sqrt{10 \cdot 30}}{- 2})}^{2}$

Multiply $10$ by $30$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 \sqrt{300}}{- 2})}^{2}$

Rewrite $300$ as $10^{2} \cdot 3$ .

Factor $100$ out of $300$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 \sqrt{100 (3)}}{- 2})}^{2}$

Rewrite $100$ as $10^{2}$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 \sqrt{10^{2} \cdot 3}}{- 2})}^{2}$

Pull terms out from under the radical.

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 2 (10 \sqrt{3})}{- 2})}^{2}$

Multiply $10$ by $- 2$ .

${(\frac{18 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30} - 20 \sqrt{3}}{- 2})}^{2}$

Subtract $20 \sqrt{3}$ from $18 \sqrt{3}$ .

${(\frac{- 2 \sqrt{3} - 6 \sqrt{30} + 6 \sqrt{30}}{- 2})}^{2}$

Add $- 6 \sqrt{30}$ and $6 \sqrt{30}$ .

${(\frac{- 2 \sqrt{3} + 0}{- 2})}^{2}$

Add $- 2 \sqrt{3}$ and $0$ .

${(\frac{- 2 \sqrt{3}}{- 2})}^{2}$

Step 9

Simplify terms.

Cancel the common factor of $- 2$ .

Cancel the common factor.

${(\frac{- 2 \sqrt{3}}{- 2})}^{2}$

Divide $\sqrt{3}$ by $1$ .

${\sqrt{3}}^{2}$

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

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

${(3^{\frac{1}{2}})}^{2}$

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

$3^{\frac{1}{2} \cdot 2}$

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

$3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3^{\frac{2}{2}}$

Rewrite the expression.

$3^{1}$

Evaluate the exponent.

$3$

Simplify (( square root of 6+ square root of 15+ square root of 3)/( fourth root of 4+ square root of 5+5^0))^2

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