Solve 3(sqrt{18}+3)-sqrt{5}(sqrt{15}-sqrt{30})

$$3 ( \sqrt { 18 } + 3 ) - \sqrt { 5 } ( \sqrt { 15 } - \sqrt { 30 } )$$

Evaluate

$$5\sqrt{6}+9\sqrt{2}+9-5\sqrt{3}\approx 25.315116737$$

Solution Steps

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}.

$$3\left(3\sqrt{2}+3\right)-\sqrt{5}\left(\sqrt{15}-\sqrt{30}\right)$$

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

$$9\sqrt{2}+9-\sqrt{5}\left(\sqrt{15}-\sqrt{30}\right)$$

Use the distributive property to multiply \sqrt{5} by \sqrt{15}-\sqrt{30}.

$$9\sqrt{2}+9-\left(\sqrt{5}\sqrt{15}-\sqrt{5}\sqrt{30}\right)$$

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}.

$$9\sqrt{2}+9-\left(\sqrt{5}\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{30}\right)$$

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

$$9\sqrt{2}+9-\left(5\sqrt{3}-\sqrt{5}\sqrt{30}\right)$$

Factor 30=5\times 6. Rewrite the square root of the product \sqrt{5\times 6} as the product of square roots \sqrt{5}\sqrt{6}.

$$9\sqrt{2}+9-\left(5\sqrt{3}-\sqrt{5}\sqrt{5}\sqrt{6}\right)$$

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

$$9\sqrt{2}+9-\left(5\sqrt{3}-5\sqrt{6}\right)$$

To find the opposite of 5\sqrt{3}-5\sqrt{6}, find the opposite of each term.

$$9\sqrt{2}+9-5\sqrt{3}-\left(-5\sqrt{6}\right)$$

The opposite of -5\sqrt{6} is 5\sqrt{6}.

$$9\sqrt{2}+9-5\sqrt{3}+5\sqrt{6}$$