Solve (sqrt{30}+sqrt{21}-3)(sqrt{3}+sqrt{10}-sqrt{7})

$$( \sqrt { 30 } + \sqrt { 21 } - 3 ) ( \sqrt { 3 } + \sqrt { 10 } - \sqrt { 7 } )$$

Evaluate

$$6\sqrt{7}\approx 15.874507866$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{30}+\sqrt{21}-3 by each term of \sqrt{3}+\sqrt{10}-\sqrt{7}.

$$\sqrt{30}\sqrt{3}+\sqrt{30}\sqrt{10}-\sqrt{30}\sqrt{7}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\sqrt{3}\sqrt{10}\sqrt{3}+\sqrt{30}\sqrt{10}-\sqrt{30}\sqrt{7}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$3\sqrt{10}+\sqrt{30}\sqrt{10}-\sqrt{30}\sqrt{7}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Factor 30=10\times 3. Rewrite the square root of the product \sqrt{10\times 3} as the product of square roots \sqrt{10}\sqrt{3}.

$$3\sqrt{10}+\sqrt{10}\sqrt{3}\sqrt{10}-\sqrt{30}\sqrt{7}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$3\sqrt{10}+10\sqrt{3}-\sqrt{30}\sqrt{7}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

To multiply \sqrt{30} and \sqrt{7}, multiply the numbers under the square root.

$$3\sqrt{10}+10\sqrt{3}-\sqrt{210}+\sqrt{21}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

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

$$3\sqrt{10}+10\sqrt{3}-\sqrt{210}+\sqrt{3}\sqrt{7}\sqrt{3}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$3\sqrt{10}+10\sqrt{3}-\sqrt{210}+3\sqrt{7}+\sqrt{21}\sqrt{10}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

To multiply \sqrt{21} and \sqrt{10}, multiply the numbers under the square root.

$$3\sqrt{10}+10\sqrt{3}-\sqrt{210}+3\sqrt{7}+\sqrt{210}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Combine -\sqrt{210} and \sqrt{210} to get 0.

$$3\sqrt{10}+10\sqrt{3}+3\sqrt{7}-\sqrt{21}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

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}.

$$3\sqrt{10}+10\sqrt{3}+3\sqrt{7}-\sqrt{7}\sqrt{3}\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

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

$$3\sqrt{10}+10\sqrt{3}+3\sqrt{7}-7\sqrt{3}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Combine 10\sqrt{3} and -7\sqrt{3} to get 3\sqrt{3}.

$$3\sqrt{10}+3\sqrt{3}+3\sqrt{7}-3\sqrt{3}-3\sqrt{10}+3\sqrt{7}$$

Combine 3\sqrt{3} and -3\sqrt{3} to get 0.

$$3\sqrt{10}+3\sqrt{7}-3\sqrt{10}+3\sqrt{7}$$

Combine 3\sqrt{10} and -3\sqrt{10} to get 0.

$$3\sqrt{7}+3\sqrt{7}$$

Combine 3\sqrt{7} and 3\sqrt{7} to get 6\sqrt{7}.

$$6\sqrt{7}$$