Solve (2sqrt{12}-5sqrt{8})-(sqrt{27}-sqrt{18})

$$(2 \sqrt{ 12 } -5 \sqrt{ 8 } )-( \sqrt{ 27 } - \sqrt{ 18 } )$$

Evaluate

$$\sqrt{3}-7\sqrt{2}\approx -8.167444129$$

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}.

$$2\times 2\sqrt{3}-5\sqrt{8}-\left(\sqrt{27}-\sqrt{18}\right)$$

Multiply 2 and 2 to get 4.

$$4\sqrt{3}-5\sqrt{8}-\left(\sqrt{27}-\sqrt{18}\right)$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$4\sqrt{3}-5\times 2\sqrt{2}-\left(\sqrt{27}-\sqrt{18}\right)$$

Multiply -5 and 2 to get -10.

$$4\sqrt{3}-10\sqrt{2}-\left(\sqrt{27}-\sqrt{18}\right)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$4\sqrt{3}-10\sqrt{2}-\left(3\sqrt{3}-\sqrt{18}\right)$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$4\sqrt{3}-10\sqrt{2}-\left(3\sqrt{3}-3\sqrt{2}\right)$$

To find the opposite of 3\sqrt{3}-3\sqrt{2}, find the opposite of each term.

$$4\sqrt{3}-10\sqrt{2}-3\sqrt{3}-\left(-3\sqrt{2}\right)$$

Combine 4\sqrt{3} and -3\sqrt{3} to get \sqrt{3}.

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

The opposite of -3\sqrt{2} is 3\sqrt{2}.

$$\sqrt{3}-10\sqrt{2}+3\sqrt{2}$$

Combine -10\sqrt{2} and 3\sqrt{2} to get -7\sqrt{2}.

$$\sqrt{3}-7\sqrt{2}$$