Solve (sqrt{5}-sqrt{7})left(sqrt{21}-sqrt{15}right)

$$( \sqrt{ 5 } - \sqrt{ 7 } ) \left( \sqrt{ 21 } - \sqrt{ 15 } \right)$$

Evaluate

$$2\left(\sqrt{105}-6\sqrt{3}\right)\approx -0.290708159$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{5}-\sqrt{7} by each term of \sqrt{21}-\sqrt{15}.

$$\sqrt{5}\sqrt{21}-\sqrt{5}\sqrt{15}-\sqrt{7}\sqrt{21}+\sqrt{15}\sqrt{7}$$

To multiply \sqrt{5} and \sqrt{21}, multiply the numbers under the square root.

$$\sqrt{105}-\sqrt{5}\sqrt{15}-\sqrt{7}\sqrt{21}+\sqrt{15}\sqrt{7}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\sqrt{105}-\sqrt{5}\sqrt{5}\sqrt{3}-\sqrt{7}\sqrt{21}+\sqrt{15}\sqrt{7}$$

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

$$\sqrt{105}-5\sqrt{3}-\sqrt{7}\sqrt{21}+\sqrt{15}\sqrt{7}$$

Factor 21=7\times 3. Rewrite the square root of the product \sqrt{7\times 3} as the product of square roots \sqrt{7}\sqrt{3}.

$$\sqrt{105}-5\sqrt{3}-\sqrt{7}\sqrt{7}\sqrt{3}+\sqrt{15}\sqrt{7}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\sqrt{105}-5\sqrt{3}-7\sqrt{3}+\sqrt{15}\sqrt{7}$$

Combine -5\sqrt{3} and -7\sqrt{3} to get -12\sqrt{3}.

$$\sqrt{105}-12\sqrt{3}+\sqrt{15}\sqrt{7}$$

To multiply \sqrt{15} and \sqrt{7}, multiply the numbers under the square root.

$$\sqrt{105}-12\sqrt{3}+\sqrt{105}$$

Combine \sqrt{105} and \sqrt{105} to get 2\sqrt{105}.

$$2\sqrt{105}-12\sqrt{3}$$