Solve sqrt{125}-(2sqrt{21}-7sqrt{75}+4sqrt{45})

$$\sqrt { 125 } - ( 2 \sqrt { 21 } - 7 \sqrt { 75 } + 4 \sqrt { 45 } )$$

Evaluate

$$35\sqrt{3}-2\sqrt{21}-7\sqrt{5}\approx 35.804151033$$

Solution Steps

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}.

$$5\sqrt{5}-\left(2\sqrt{21}-7\sqrt{75}+4\sqrt{45}\right)$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$5\sqrt{5}-\left(2\sqrt{21}-7\times 5\sqrt{3}+4\sqrt{45}\right)$$

Multiply -7 and 5 to get -35.

$$5\sqrt{5}-\left(2\sqrt{21}-35\sqrt{3}+4\sqrt{45}\right)$$

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}.

$$5\sqrt{5}-\left(2\sqrt{21}-35\sqrt{3}+4\times 3\sqrt{5}\right)$$

Multiply 4 and 3 to get 12.

$$5\sqrt{5}-\left(2\sqrt{21}-35\sqrt{3}+12\sqrt{5}\right)$$

To find the opposite of 2\sqrt{21}-35\sqrt{3}+12\sqrt{5}, find the opposite of each term.

$$5\sqrt{5}-2\sqrt{21}-\left(-35\sqrt{3}\right)-12\sqrt{5}$$

The opposite of -35\sqrt{3} is 35\sqrt{3}.

$$5\sqrt{5}-2\sqrt{21}+35\sqrt{3}-12\sqrt{5}$$

Combine 5\sqrt{5} and -12\sqrt{5} to get -7\sqrt{5}.

$$-7\sqrt{5}-2\sqrt{21}+35\sqrt{3}$$