Solve frac{sqrt{18}-sqrt{12}}{sqrt{50}-sqrt{48}}

$$\frac { \sqrt { 18 } - \sqrt { 12 } } { \sqrt { 50 } - \sqrt { 48 } }$$

Evaluate

$$\sqrt{6}+3\approx 5.449489743$$

Solution Steps

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}.

$$\frac{3\sqrt{2}-\sqrt{12}}{\sqrt{50}-\sqrt{48}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{3\sqrt{2}-2\sqrt{3}}{\sqrt{50}-\sqrt{48}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{3\sqrt{2}-2\sqrt{3}}{5\sqrt{2}-\sqrt{48}}$$

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}.

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

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

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

Consider \left(5\sqrt{2}-4\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(5\sqrt{2}\right)^{2}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\left(\sqrt{2}\right)^{2}-\left(-4\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{25\times 2-\left(-4\sqrt{3}\right)^{2}}$$

Multiply 25 and 2 to get 50.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{\left(3\sqrt{2}-2\sqrt{3}\right)\left(5\sqrt{2}+4\sqrt{3}\right)}{50-48}$$

Subtract 48 from 50 to get 2.

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

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

$$\frac{15\left(\sqrt{2}\right)^{2}+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{15\times 2+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

Multiply 15 and 2 to get 30.

$$\frac{30+12\sqrt{3}\sqrt{2}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

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

$$\frac{30+12\sqrt{6}-10\sqrt{3}\sqrt{2}-8\left(\sqrt{3}\right)^{2}}{2}$$

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

$$\frac{30+12\sqrt{6}-10\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

Combine 12\sqrt{6} and -10\sqrt{6} to get 2\sqrt{6}.

$$\frac{30+2\sqrt{6}-8\left(\sqrt{3}\right)^{2}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{30+2\sqrt{6}-8\times 3}{2}$$

Multiply -8 and 3 to get -24.

$$\frac{30+2\sqrt{6}-24}{2}$$

Subtract 24 from 30 to get 6.

$$\frac{6+2\sqrt{6}}{2}$$

Divide each term of 6+2\sqrt{6} by 2 to get 3+\sqrt{6}.

$$3+\sqrt{6}$$