Simplify (5 square root of 50-3 square root of 200+3 square root of 18)^2

${(5 \sqrt{50} - 3 \sqrt{200} + 3 \sqrt{18})}^{2}$

Step 1

Simplify each term.

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

${(5 \sqrt{25 (2)} - 3 \sqrt{200} + 3 \sqrt{18})}^{2}$

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

${(5 \sqrt{5^{2} \cdot 2} - 3 \sqrt{200} + 3 \sqrt{18})}^{2}$

Pull terms out from under the radical.

${(5 (5 \sqrt{2}) - 3 \sqrt{200} + 3 \sqrt{18})}^{2}$

Multiply $5$ by $5$ .

${(25 \sqrt{2} - 3 \sqrt{200} + 3 \sqrt{18})}^{2}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

${(25 \sqrt{2} - 3 \sqrt{100 (2)} + 3 \sqrt{18})}^{2}$

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

${(25 \sqrt{2} - 3 \sqrt{10^{2} \cdot 2} + 3 \sqrt{18})}^{2}$

Pull terms out from under the radical.

${(25 \sqrt{2} - 3 (10 \sqrt{2}) + 3 \sqrt{18})}^{2}$

Multiply $10$ by $- 3$ .

${(25 \sqrt{2} - 30 \sqrt{2} + 3 \sqrt{18})}^{2}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

${(25 \sqrt{2} - 30 \sqrt{2} + 3 \sqrt{9 (2)})}^{2}$

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

${(25 \sqrt{2} - 30 \sqrt{2} + 3 \sqrt{3^{2} \cdot 2})}^{2}$

Pull terms out from under the radical.

${(25 \sqrt{2} - 30 \sqrt{2} + 3 (3 \sqrt{2}))}^{2}$

Multiply $3$ by $3$ .

${(25 \sqrt{2} - 30 \sqrt{2} + 9 \sqrt{2})}^{2}$

Step 2

Simplify terms.

Subtract $30 \sqrt{2}$ from $25 \sqrt{2}$ .

${(- 5 \sqrt{2} + 9 \sqrt{2})}^{2}$

Add $- 5 \sqrt{2}$ and $9 \sqrt{2}$ .

${(4 \sqrt{2})}^{2}$

Simplify the expression.

Apply the product rule to $4 \sqrt{2}$ .

$4^{2} {\sqrt{2}}^{2}$

Raise $4$ to the power of $2$ .

$16 {\sqrt{2}}^{2}$

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

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

$16 {(2^{\frac{1}{2}})}^{2}$

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

$16 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$16 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$16 \cdot 2^{1}$

Evaluate the exponent.

$16 \cdot 2$

Multiply $16$ by $2$ .

$32$

Simplify (5 square root of 50-3 square root of 200+3 square root of 18)^2

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