Solve (sqrt{3}+45/sqrt{3})*(4sqrt{3}-45/4sqrt{3})

$$( \sqrt{ 3 } + \frac{ 45 }{ \sqrt{ 3 } } ) \times (4 \sqrt{ 3 } - \frac{ 45 }{ 4 \sqrt{ 3 } } )$$

Evaluate

$$12$$

Solution Steps

Rationalize the denominator of \frac{45}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(\sqrt{3}+\frac{45\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)\left(4\sqrt{3}-\frac{45}{4\sqrt{3}}\right)$$

The square of \sqrt{3} is 3.

$$\left(\sqrt{3}+\frac{45\sqrt{3}}{3}\right)\left(4\sqrt{3}-\frac{45}{4\sqrt{3}}\right)$$

Divide 45\sqrt{3} by 3 to get 15\sqrt{3}.

$$\left(\sqrt{3}+15\sqrt{3}\right)\left(4\sqrt{3}-\frac{45}{4\sqrt{3}}\right)$$

Combine \sqrt{3} and 15\sqrt{3} to get 16\sqrt{3}.

$$16\sqrt{3}\left(4\sqrt{3}-\frac{45}{4\sqrt{3}}\right)$$

Rationalize the denominator of \frac{45}{4\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$16\sqrt{3}\left(4\sqrt{3}-\frac{45\sqrt{3}}{4\left(\sqrt{3}\right)^{2}}\right)$$

The square of \sqrt{3} is 3.

$$16\sqrt{3}\left(4\sqrt{3}-\frac{45\sqrt{3}}{4\times 3}\right)$$

Cancel out 3 in both numerator and denominator.

$$16\sqrt{3}\left(4\sqrt{3}-\frac{15\sqrt{3}}{4}\right)$$

Combine 4\sqrt{3} and -\frac{15\sqrt{3}}{4} to get \frac{1}{4}\sqrt{3}.

$$16\sqrt{3}\times \frac{1}{4}\sqrt{3}$$

Multiply 16 and \frac{1}{4} to get \frac{16}{4}.

$$\frac{16}{4}\sqrt{3}\sqrt{3}$$

Divide 16 by 4 to get 4.

$$4\sqrt{3}\sqrt{3}$$

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

$$4\times 3$$

Multiply 4 and 3 to get 12.

$$12$$

Factor

$$2^{2}\times 3$$