Solve frac{sqrt{80}-sqrt{112}}{sqrt{45}-sqrt{63}}=

$$\frac { \sqrt { 80 } - \sqrt { 112 } } { \sqrt { 45 } - \sqrt { 63 } } =$$

Evaluate

$$\frac{4}{3}\approx 1.333333333$$

Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{4\sqrt{5}-\sqrt{112}}{\sqrt{45}-\sqrt{63}}$$

Factor 112=4^{2}\times 7. Rewrite the square root of the product \sqrt{4^{2}\times 7} as the product of square roots \sqrt{4^{2}}\sqrt{7}. Take the square root of 4^{2}.

$$\frac{4\sqrt{5}-4\sqrt{7}}{\sqrt{45}-\sqrt{63}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{4\sqrt{5}-4\sqrt{7}}{3\sqrt{5}-\sqrt{63}}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

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

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

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

Consider \left(3\sqrt{5}-3\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3\sqrt{5}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

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

Multiply 9 and 5 to get 45.

$$\frac{\left(4\sqrt{5}-4\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right)}{45-\left(-3\sqrt{7}\right)^{2}}$$

Expand \left(-3\sqrt{7}\right)^{2}.

$$\frac{\left(4\sqrt{5}-4\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right)}{45-\left(-3\right)^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(4\sqrt{5}-4\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right)}{45-9\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(4\sqrt{5}-4\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right)}{45-9\times 7}$$

Multiply 9 and 7 to get 63.

$$\frac{\left(4\sqrt{5}-4\sqrt{7}\right)\left(3\sqrt{5}+3\sqrt{7}\right)}{45-63}$$

Subtract 63 from 45 to get -18.

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

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

$$\frac{12\left(\sqrt{5}\right)^{2}+12\sqrt{5}\sqrt{7}-12\sqrt{7}\sqrt{5}-12\left(\sqrt{7}\right)^{2}}{-18}$$

The square of \sqrt{5} is 5.

$$\frac{12\times 5+12\sqrt{5}\sqrt{7}-12\sqrt{7}\sqrt{5}-12\left(\sqrt{7}\right)^{2}}{-18}$$

Multiply 12 and 5 to get 60.

$$\frac{60+12\sqrt{5}\sqrt{7}-12\sqrt{7}\sqrt{5}-12\left(\sqrt{7}\right)^{2}}{-18}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{60+12\sqrt{35}-12\sqrt{7}\sqrt{5}-12\left(\sqrt{7}\right)^{2}}{-18}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{60+12\sqrt{35}-12\sqrt{35}-12\left(\sqrt{7}\right)^{2}}{-18}$$

Combine 12\sqrt{35} and -12\sqrt{35} to get 0.

$$\frac{60-12\left(\sqrt{7}\right)^{2}}{-18}$$

The square of \sqrt{7} is 7.

$$\frac{60-12\times 7}{-18}$$

Multiply -12 and 7 to get -84.

$$\frac{60-84}{-18}$$

Subtract 84 from 60 to get -24.

$$\frac{-24}{-18}$$

Reduce the fraction \frac{-24}{-18} to lowest terms by extracting and canceling out -6.

$$\frac{4}{3}$$

Factor

$$\frac{2 ^ {2}}{3} = 1\frac{1}{3} = 1.3333333333333333$$