Solve (sqrt{48}-sqrt{50}+sqrt{48}-sqrt{45})-(sqrt{50}-sqrt{45})^2

$$( \sqrt { 48 } - \sqrt { 50 } + \sqrt { 48 } - \sqrt { 45 } ) - ( \sqrt { 50 } - \sqrt { 45 } ) ^ { 2 }$$

Evaluate

$$8\sqrt{3}+30\sqrt{10}-3\sqrt{5}-5\sqrt{2}-95\approx -0.054535479$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$4\sqrt{3}-\sqrt{50}+\sqrt{48}-\sqrt{45}-\left(\sqrt{50}-\sqrt{45}\right)^{2}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$4\sqrt{3}-5\sqrt{2}+\sqrt{48}-\sqrt{45}-\left(\sqrt{50}-\sqrt{45}\right)^{2}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$4\sqrt{3}-5\sqrt{2}+4\sqrt{3}-\sqrt{45}-\left(\sqrt{50}-\sqrt{45}\right)^{2}$$

Combine 4\sqrt{3} and 4\sqrt{3} to get 8\sqrt{3}.

$$8\sqrt{3}-5\sqrt{2}-\sqrt{45}-\left(\sqrt{50}-\sqrt{45}\right)^{2}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(\sqrt{50}-\sqrt{45}\right)^{2}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(5\sqrt{2}-\sqrt{45}\right)^{2}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(5\sqrt{2}-3\sqrt{5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5\sqrt{2}-3\sqrt{5}\right)^{2}.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(25\left(\sqrt{2}\right)^{2}-30\sqrt{2}\sqrt{5}+9\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(25\times 2-30\sqrt{2}\sqrt{5}+9\left(\sqrt{5}\right)^{2}\right)$$

Multiply 25 and 2 to get 50.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(50-30\sqrt{2}\sqrt{5}+9\left(\sqrt{5}\right)^{2}\right)$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(50-30\sqrt{10}+9\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(50-30\sqrt{10}+9\times 5\right)$$

Multiply 9 and 5 to get 45.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(50-30\sqrt{10}+45\right)$$

Add 50 and 45 to get 95.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-\left(95-30\sqrt{10}\right)$$

To find the opposite of 95-30\sqrt{10}, find the opposite of each term.

$$8\sqrt{3}-5\sqrt{2}-3\sqrt{5}-95+30\sqrt{10}$$