Solve (sqrt{45}+sqrt{18})-(sqrt{8}-sqrt{125})

$$( \sqrt{ 45 } + \sqrt{ 18 } )-( \sqrt{ 8 } - \sqrt{ 125 } )$$

Evaluate

$$\sqrt{2}+8\sqrt{5}\approx 19.302757382$$

Solution Steps

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}.

$$3\sqrt{5}+\sqrt{18}-\left(\sqrt{8}-\sqrt{125}\right)$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{5}+3\sqrt{2}-\left(\sqrt{8}-\sqrt{125}\right)$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$3\sqrt{5}+3\sqrt{2}-\left(2\sqrt{2}-\sqrt{125}\right)$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

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

To find the opposite of 2\sqrt{2}-5\sqrt{5}, find the opposite of each term.

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

Combine 3\sqrt{2} and -2\sqrt{2} to get \sqrt{2}.

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

The opposite of -5\sqrt{5} is 5\sqrt{5}.

$$3\sqrt{5}+\sqrt{2}+5\sqrt{5}$$

Combine 3\sqrt{5} and 5\sqrt{5} to get 8\sqrt{5}.

$$8\sqrt{5}+\sqrt{2}$$