Solve =frac{7sqrt{3}-5sqrt{2}}{sqrt{48}+sqrt{18}}

$$= \frac { 7 \sqrt { 3 } - 5 \sqrt { 2 } } { \sqrt { 48 } + \sqrt { 18 } }$$

Evaluate

$$-\frac{41\sqrt{6}}{30}+\frac{19}{5}\approx 0.452364018$$

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}.

$$\frac{7\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+\sqrt{18}}$$

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}.

$$\frac{7\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}$$

Rationalize the denominator of \frac{7\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}} by multiplying numerator and denominator by 4\sqrt{3}-3\sqrt{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{\left(4\sqrt{3}+3\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}$$

Consider \left(4\sqrt{3}+3\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{\left(4\sqrt{3}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

Expand \left(4\sqrt{3}\right)^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{4^{2}\left(\sqrt{3}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{16\left(\sqrt{3}\right)^{2}-\left(3\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{16\times 3-\left(3\sqrt{2}\right)^{2}}$$

Multiply 16 and 3 to get 48.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-\left(3\sqrt{2}\right)^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-3^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{48-18}$$

Subtract 18 from 48 to get 30.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-3\sqrt{2}\right)}{30}$$

Apply the distributive property by multiplying each term of 7\sqrt{3}-5\sqrt{2} by each term of 4\sqrt{3}-3\sqrt{2}.

$$\frac{28\left(\sqrt{3}\right)^{2}-21\sqrt{3}\sqrt{2}-20\sqrt{3}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{30}$$

The square of \sqrt{3} is 3.

$$\frac{28\times 3-21\sqrt{3}\sqrt{2}-20\sqrt{3}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{30}$$

Multiply 28 and 3 to get 84.

$$\frac{84-21\sqrt{3}\sqrt{2}-20\sqrt{3}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{30}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{84-21\sqrt{6}-20\sqrt{3}\sqrt{2}+15\left(\sqrt{2}\right)^{2}}{30}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{84-21\sqrt{6}-20\sqrt{6}+15\left(\sqrt{2}\right)^{2}}{30}$$

Combine -21\sqrt{6} and -20\sqrt{6} to get -41\sqrt{6}.

$$\frac{84-41\sqrt{6}+15\left(\sqrt{2}\right)^{2}}{30}$$

The square of \sqrt{2} is 2.

$$\frac{84-41\sqrt{6}+15\times 2}{30}$$

Multiply 15 and 2 to get 30.

$$\frac{84-41\sqrt{6}+30}{30}$$

Add 84 and 30 to get 114.

$$\frac{114-41\sqrt{6}}{30}$$