Solve frac{sqrt{8}}{5*sqrt{8}+14*(-sqrt{2})^2}

$$\frac { \sqrt { 8 } } { 5 \cdot \sqrt { 8 } + 14 \cdot ( - \sqrt { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{7\sqrt{2}-5}{73}\approx 0.067116369$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{5\sqrt{8}+14\left(-\sqrt{2}\right)^{2}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{5\times 2\sqrt{2}+14\left(-\sqrt{2}\right)^{2}}$$

Multiply 5 and 2 to get 10.

$$\frac{2\sqrt{2}}{10\sqrt{2}+14\left(-\sqrt{2}\right)^{2}}$$

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

$$\frac{2\sqrt{2}}{10\sqrt{2}+14\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}}{10\sqrt{2}+14\times 2}$$

Multiply 14 and 2 to get 28.

$$\frac{2\sqrt{2}}{10\sqrt{2}+28}$$

Rationalize the denominator of \frac{2\sqrt{2}}{10\sqrt{2}+28} by multiplying numerator and denominator by 10\sqrt{2}-28.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{\left(10\sqrt{2}+28\right)\left(10\sqrt{2}-28\right)}$$

Consider \left(10\sqrt{2}+28\right)\left(10\sqrt{2}-28\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{\left(10\sqrt{2}\right)^{2}-28^{2}}$$

Expand \left(10\sqrt{2}\right)^{2}.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{10^{2}\left(\sqrt{2}\right)^{2}-28^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{100\left(\sqrt{2}\right)^{2}-28^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{100\times 2-28^{2}}$$

Multiply 100 and 2 to get 200.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{200-28^{2}}$$

Calculate 28 to the power of 2 and get 784.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{200-784}$$

Subtract 784 from 200 to get -584.

$$\frac{2\sqrt{2}\left(10\sqrt{2}-28\right)}{-584}$$

Divide 2\sqrt{2}\left(10\sqrt{2}-28\right) by -584 to get -\frac{1}{292}\sqrt{2}\left(10\sqrt{2}-28\right).

$$-\frac{1}{292}\sqrt{2}\left(10\sqrt{2}-28\right)$$

Use the distributive property to multiply -\frac{1}{292}\sqrt{2} by 10\sqrt{2}-28.

$$-\frac{5}{146}\left(\sqrt{2}\right)^{2}+\frac{7}{73}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$-\frac{5}{146}\times 2+\frac{7}{73}\sqrt{2}$$

Multiply -\frac{5}{146} and 2 to get -\frac{5}{73}.

$$-\frac{5}{73}+\frac{7}{73}\sqrt{2}$$