Solve (sqrt{10}-7sqrt{5})(3sqrt{10}+2sqrt{5})

$$( \sqrt { 10 } - 7 \sqrt { 5 } ) ( 3 \sqrt { 10 } + 2 \sqrt { 5 } )$$

Evaluate

$$-95\sqrt{2}-40\approx -174.350288425$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{10}-7\sqrt{5} by each term of 3\sqrt{10}+2\sqrt{5}.

$$3\left(\sqrt{10}\right)^{2}+2\sqrt{10}\sqrt{5}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{10} is 10.

$$3\times 10+2\sqrt{10}\sqrt{5}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

Multiply 3 and 10 to get 30.

$$30+2\sqrt{10}\sqrt{5}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

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}.

$$30+2\sqrt{5}\sqrt{2}\sqrt{5}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

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

$$30+2\times 5\sqrt{2}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

Multiply 2 and 5 to get 10.

$$30+10\sqrt{2}-21\sqrt{5}\sqrt{10}-14\left(\sqrt{5}\right)^{2}$$

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}.

$$30+10\sqrt{2}-21\sqrt{5}\sqrt{5}\sqrt{2}-14\left(\sqrt{5}\right)^{2}$$

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

$$30+10\sqrt{2}-21\times 5\sqrt{2}-14\left(\sqrt{5}\right)^{2}$$

Multiply -21 and 5 to get -105.

$$30+10\sqrt{2}-105\sqrt{2}-14\left(\sqrt{5}\right)^{2}$$

Combine 10\sqrt{2} and -105\sqrt{2} to get -95\sqrt{2}.

$$30-95\sqrt{2}-14\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$30-95\sqrt{2}-14\times 5$$

Multiply -14 and 5 to get -70.

$$30-95\sqrt{2}-70$$

Subtract 70 from 30 to get -40.

$$-40-95\sqrt{2}$$