Solve frac{24sqrt{2}+36}{27sqrt{2}+36}

$$\frac { 24 \sqrt { 2 } + 36 } { 27 \sqrt { 2 } + 36 }$$

Evaluate

$$\frac{2\sqrt{2}}{3}\approx 0.942809042$$

Solution Steps

Rationalize the denominator of \frac{24\sqrt{2}+36}{27\sqrt{2}+36} by multiplying numerator and denominator by 27\sqrt{2}-36.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{\left(27\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}$$

Consider \left(27\sqrt{2}+36\right)\left(27\sqrt{2}-36\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{\left(27\sqrt{2}\right)^{2}-36^{2}}$$

Expand \left(27\sqrt{2}\right)^{2}.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{27^{2}\left(\sqrt{2}\right)^{2}-36^{2}}$$

Calculate 27 to the power of 2 and get 729.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{729\left(\sqrt{2}\right)^{2}-36^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{729\times 2-36^{2}}$$

Multiply 729 and 2 to get 1458.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{1458-36^{2}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{1458-1296}$$

Subtract 1296 from 1458 to get 162.

$$\frac{\left(24\sqrt{2}+36\right)\left(27\sqrt{2}-36\right)}{162}$$

Apply the distributive property by multiplying each term of 24\sqrt{2}+36 by each term of 27\sqrt{2}-36.

$$\frac{648\left(\sqrt{2}\right)^{2}-864\sqrt{2}+972\sqrt{2}-1296}{162}$$

The square of \sqrt{2} is 2.

$$\frac{648\times 2-864\sqrt{2}+972\sqrt{2}-1296}{162}$$

Multiply 648 and 2 to get 1296.

$$\frac{1296-864\sqrt{2}+972\sqrt{2}-1296}{162}$$

Combine -864\sqrt{2} and 972\sqrt{2} to get 108\sqrt{2}.

$$\frac{1296+108\sqrt{2}-1296}{162}$$

Subtract 1296 from 1296 to get 0.

$$\frac{108\sqrt{2}}{162}$$

Divide 108\sqrt{2} by 162 to get \frac{2}{3}\sqrt{2}.

$$\frac{2}{3}\sqrt{2}$$