Solve (sqrt{10}-6sqrt{5})(5sqrt{10}+5sqrt{5})

$$( \sqrt { 10 } - 6 \sqrt { 5 } ) ( 5 \sqrt { 10 } + 5 \sqrt { 5 } )$$

Evaluate

$$-125\sqrt{2}-100\approx -276.776695297$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{10}-6\sqrt{5} by each term of 5\sqrt{10}+5\sqrt{5}.

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

The square of \sqrt{10} is 10.

$$5\times 10+5\sqrt{10}\sqrt{5}-30\sqrt{5}\sqrt{10}-30\left(\sqrt{5}\right)^{2}$$

Multiply 5 and 10 to get 50.

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

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply 5 and 5 to get 25.

$$50+25\sqrt{2}-30\sqrt{5}\sqrt{10}-30\left(\sqrt{5}\right)^{2}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$50+25\sqrt{2}-30\sqrt{5}\sqrt{5}\sqrt{2}-30\left(\sqrt{5}\right)^{2}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$50+25\sqrt{2}-30\times 5\sqrt{2}-30\left(\sqrt{5}\right)^{2}$$

Multiply -30 and 5 to get -150.

$$50+25\sqrt{2}-150\sqrt{2}-30\left(\sqrt{5}\right)^{2}$$

Combine 25\sqrt{2} and -150\sqrt{2} to get -125\sqrt{2}.

$$50-125\sqrt{2}-30\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$50-125\sqrt{2}-30\times 5$$

Multiply -30 and 5 to get -150.

$$50-125\sqrt{2}-150$$

Subtract 150 from 50 to get -100.

$$-100-125\sqrt{2}$$

Factor

$$25 {(-5 \sqrt{2} - 4)} = -276.776695297$$