Solve frac{left(7sqrt{3}-5sqrt{2}right)left(sqrt{48}-sqrt{8}right)}{{left(sqrt{48}right)}^2-{left(sqrt{8}right)}^2}

$$\frac{ \left( 7 \sqrt{ 3 } -5 \sqrt{ 2 } \right) \left( \sqrt{ 48 } - \sqrt{ 8 } \right) }{ { \left( \sqrt{ 48 } \right) }^{ 2 } - { \left( \sqrt{ 8 } \right) }^{ 2 } }$$

Evaluate

$$-\frac{17\sqrt{6}}{20}+\frac{13}{5}\approx 0.517933719$$

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}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-\sqrt{8}\right)}{\left(\sqrt{48}\right)^{2}-\left(\sqrt{8}\right)^{2}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-2\sqrt{2}\right)}{\left(\sqrt{48}\right)^{2}-\left(\sqrt{8}\right)^{2}}$$

The square of \sqrt{48} is 48.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-2\sqrt{2}\right)}{48-\left(\sqrt{8}\right)^{2}}$$

The square of \sqrt{8} is 8.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-2\sqrt{2}\right)}{48-8}$$

Subtract 8 from 48 to get 40.

$$\frac{\left(7\sqrt{3}-5\sqrt{2}\right)\left(4\sqrt{3}-2\sqrt{2}\right)}{40}$$

Use the distributive property to multiply 7\sqrt{3}-5\sqrt{2} by 4\sqrt{3}-2\sqrt{2} and combine like terms.

$$\frac{28\left(\sqrt{3}\right)^{2}-34\sqrt{3}\sqrt{2}+10\left(\sqrt{2}\right)^{2}}{40}$$

The square of \sqrt{3} is 3.

$$\frac{28\times 3-34\sqrt{3}\sqrt{2}+10\left(\sqrt{2}\right)^{2}}{40}$$

Multiply 28 and 3 to get 84.

$$\frac{84-34\sqrt{3}\sqrt{2}+10\left(\sqrt{2}\right)^{2}}{40}$$

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

$$\frac{84-34\sqrt{6}+10\left(\sqrt{2}\right)^{2}}{40}$$

The square of \sqrt{2} is 2.

$$\frac{84-34\sqrt{6}+10\times 2}{40}$$

Multiply 10 and 2 to get 20.

$$\frac{84-34\sqrt{6}+20}{40}$$

Add 84 and 20 to get 104.

$$\frac{104-34\sqrt{6}}{40}$$