Solve (5+sqrt{6})*(sqrt{25*3}-sqrt{18})

$$(5+ \sqrt{ 6 } ) \times ( \sqrt{ 25 \times 3 } - \sqrt{ 18 } )$$

Evaluate

$$19\sqrt{3}\approx 32.908965344$$

Solution Steps

Multiply 25 and 3 to get 75.

$$\left(5+\sqrt{6}\right)\left(\sqrt{75}-\sqrt{18}\right)$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\left(5+\sqrt{6}\right)\left(5\sqrt{3}-\sqrt{18}\right)$$

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}.

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

Apply the distributive property by multiplying each term of 5+\sqrt{6} by each term of 5\sqrt{3}-3\sqrt{2}.

$$25\sqrt{3}-15\sqrt{2}+5\sqrt{6}\sqrt{3}-3\sqrt{6}\sqrt{2}$$

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}.

$$25\sqrt{3}-15\sqrt{2}+5\sqrt{3}\sqrt{2}\sqrt{3}-3\sqrt{6}\sqrt{2}$$

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

$$25\sqrt{3}-15\sqrt{2}+5\times 3\sqrt{2}-3\sqrt{6}\sqrt{2}$$

Multiply 5 and 3 to get 15.

$$25\sqrt{3}-15\sqrt{2}+15\sqrt{2}-3\sqrt{6}\sqrt{2}$$

Combine -15\sqrt{2} and 15\sqrt{2} to get 0.

$$25\sqrt{3}-3\sqrt{6}\sqrt{2}$$

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

$$25\sqrt{3}-3\sqrt{2}\sqrt{3}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$25\sqrt{3}-3\times 2\sqrt{3}$$

Multiply -3 and 2 to get -6.

$$25\sqrt{3}-6\sqrt{3}$$

Combine 25\sqrt{3} and -6\sqrt{3} to get 19\sqrt{3}.

$$19\sqrt{3}$$