Solve frac{sqrt{100}(sqrt{81}-sqrt{49})}{sqrt{36}-sqrt{18}}

$$\frac { \sqrt { 100 } ( \sqrt { 81 } - \sqrt { 49 } ) } { \sqrt { 36 } - \sqrt { 18 } }$$

Evaluate

$$\frac{10\sqrt{2}+20}{3}\approx 11.380711875$$

Solution Steps

Calculate the square root of 100 and get 10.

$$\frac{10\left(\sqrt{81}-\sqrt{49}\right)}{\sqrt{36}-\sqrt{18}}$$

Calculate the square root of 81 and get 9.

$$\frac{10\left(9-\sqrt{49}\right)}{\sqrt{36}-\sqrt{18}}$$

Calculate the square root of 49 and get 7.

$$\frac{10\left(9-7\right)}{\sqrt{36}-\sqrt{18}}$$

Subtract 7 from 9 to get 2.

$$\frac{10\times 2}{\sqrt{36}-\sqrt{18}}$$

Multiply 10 and 2 to get 20.

$$\frac{20}{\sqrt{36}-\sqrt{18}}$$

Calculate the square root of 36 and get 6.

$$\frac{20}{6-\sqrt{18}}$$

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}.

$$\frac{20}{6-3\sqrt{2}}$$

Rationalize the denominator of \frac{20}{6-3\sqrt{2}} by multiplying numerator and denominator by 6+3\sqrt{2}.

$$\frac{20\left(6+3\sqrt{2}\right)}{\left(6-3\sqrt{2}\right)\left(6+3\sqrt{2}\right)}$$

Consider \left(6-3\sqrt{2}\right)\left(6+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}.

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

Calculate 6 to the power of 2 and get 36.

$$\frac{20\left(6+3\sqrt{2}\right)}{36-\left(-3\sqrt{2}\right)^{2}}$$

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

$$\frac{20\left(6+3\sqrt{2}\right)}{36-\left(-3\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{20\left(6+3\sqrt{2}\right)}{36-9\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{20\left(6+3\sqrt{2}\right)}{36-9\times 2}$$

Multiply 9 and 2 to get 18.

$$\frac{20\left(6+3\sqrt{2}\right)}{36-18}$$

Subtract 18 from 36 to get 18.

$$\frac{20\left(6+3\sqrt{2}\right)}{18}$$

Divide 20\left(6+3\sqrt{2}\right) by 18 to get \frac{10}{9}\left(6+3\sqrt{2}\right).

$$\frac{10}{9}\left(6+3\sqrt{2}\right)$$

Use the distributive property to multiply \frac{10}{9} by 6+3\sqrt{2}.

$$\frac{10}{9}\times 6+\frac{10}{9}\times 3\sqrt{2}$$

Express \frac{10}{9}\times 6 as a single fraction.

$$\frac{10\times 6}{9}+\frac{10}{9}\times 3\sqrt{2}$$

Multiply 10 and 6 to get 60.

$$\frac{60}{9}+\frac{10}{9}\times 3\sqrt{2}$$

Reduce the fraction \frac{60}{9} to lowest terms by extracting and canceling out 3.

$$\frac{20}{3}+\frac{10}{9}\times 3\sqrt{2}$$

Express \frac{10}{9}\times 3 as a single fraction.

$$\frac{20}{3}+\frac{10\times 3}{9}\sqrt{2}$$

Multiply 10 and 3 to get 30.

$$\frac{20}{3}+\frac{30}{9}\sqrt{2}$$

Reduce the fraction \frac{30}{9} to lowest terms by extracting and canceling out 3.

$$\frac{20}{3}+\frac{10}{3}\sqrt{2}$$