Solve (sqrt{3}+sqrt{12})(sqrt{10}+sqrt{24})=

$$( \sqrt { 3 } + \sqrt { 12 } ) ( \sqrt { 10 } + \sqrt { 24 } ) =$$

Evaluate

$$3\left(\sqrt{30}+6\sqrt{2}\right)\approx 41.887520848$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\left(\sqrt{3}+2\sqrt{3}\right)\left(\sqrt{10}+\sqrt{24}\right)$$

Combine \sqrt{3} and 2\sqrt{3} to get 3\sqrt{3}.

$$3\sqrt{3}\left(\sqrt{10}+\sqrt{24}\right)$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$3\sqrt{3}\left(\sqrt{10}+2\sqrt{6}\right)$$

Use the distributive property to multiply 3\sqrt{3} by \sqrt{10}+2\sqrt{6}.

$$3\sqrt{3}\sqrt{10}+6\sqrt{3}\sqrt{6}$$

To multiply \sqrt{3} and \sqrt{10}, multiply the numbers under the square root.

$$3\sqrt{30}+6\sqrt{3}\sqrt{6}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$3\sqrt{30}+6\sqrt{3}\sqrt{3}\sqrt{2}$$

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

$$3\sqrt{30}+6\times 3\sqrt{2}$$

Multiply 6 and 3 to get 18.

$$3\sqrt{30}+18\sqrt{2}$$