Solve frac{3sqrt{2}+4}{5sqrt{2}+7}

$$\frac{ 3 \sqrt{ 2 } +4 }{ 5 \sqrt{ 2 } +7 }$$

Evaluate

$$2-\sqrt{2}\approx 0.585786438$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{2}+4}{5\sqrt{2}+7} by multiplying numerator and denominator by 5\sqrt{2}-7.

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

Consider \left(5\sqrt{2}+7\right)\left(5\sqrt{2}-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}.

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

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

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(3\sqrt{2}+4\right)\left(5\sqrt{2}-7\right)}{25\left(\sqrt{2}\right)^{2}-7^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{2}+4\right)\left(5\sqrt{2}-7\right)}{25\times 2-7^{2}}$$

Multiply 25 and 2 to get 50.

$$\frac{\left(3\sqrt{2}+4\right)\left(5\sqrt{2}-7\right)}{50-7^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(3\sqrt{2}+4\right)\left(5\sqrt{2}-7\right)}{50-49}$$

Subtract 49 from 50 to get 1.

$$\frac{\left(3\sqrt{2}+4\right)\left(5\sqrt{2}-7\right)}{1}$$

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 3\sqrt{2}+4 by each term of 5\sqrt{2}-7.

$$15\left(\sqrt{2}\right)^{2}-21\sqrt{2}+20\sqrt{2}-28$$

The square of \sqrt{2} is 2.

$$15\times 2-21\sqrt{2}+20\sqrt{2}-28$$

Multiply 15 and 2 to get 30.

$$30-21\sqrt{2}+20\sqrt{2}-28$$

Combine -21\sqrt{2} and 20\sqrt{2} to get -\sqrt{2}.

$$30-\sqrt{2}-28$$

Subtract 28 from 30 to get 2.

$$2-\sqrt{2}$$