Solve frac{left(7sqrt{3}-2sqrt{2}right)left(2sqrt{3}-11sqrt{48}right)}{sqrt{3}}

$$\frac{ \left( 7 \sqrt{ 3 } -2 \sqrt{ 2 } \right) \left( 2 \sqrt{ 3 } -11 \sqrt{ 48 } \right) }{ \sqrt{ 3 } }$$

Evaluate

$$42\left(2\sqrt{2}-7\sqrt{3}\right)\approx -390.428998186$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(2\sqrt{3}-11\times 4\sqrt{3}\right)}{\sqrt{3}}$$

Multiply -11 and 4 to get -44.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(2\sqrt{3}-44\sqrt{3}\right)}{\sqrt{3}}$$

Combine 2\sqrt{3} and -44\sqrt{3} to get -42\sqrt{3}.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)\sqrt{3}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)\sqrt{3}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)\sqrt{3}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)\sqrt{3}\sqrt{3}}{3}$$

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

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)\times 3}{3}$$

Multiply -42 and 3 to get -126.

$$\frac{\left(7\sqrt{3}-2\sqrt{2}\right)\left(-126\right)}{3}$$

Divide \left(7\sqrt{3}-2\sqrt{2}\right)\left(-126\right) by 3 to get \left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right).

$$\left(7\sqrt{3}-2\sqrt{2}\right)\left(-42\right)$$

Use the distributive property to multiply 7\sqrt{3}-2\sqrt{2} by -42.

$$-294\sqrt{3}+84\sqrt{2}$$