Solve frac{2left(0.8+sqrt{2}right)}{{left(0.2sqrt{2}-0.06right)}^2}

$$\frac{ 2 \left( 0.8+ \sqrt{ 2 } \right) }{ { \left(0.2 \sqrt{ 2 } -0.06 \right) }^{ 2 } }$$

Evaluate

$$\frac{1285000\sqrt{2}+1436000}{36481}\approx 89.176953144$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.2\sqrt{2}-0.06\right)^{2}.

$$\frac{2\left(0.8+\sqrt{2}\right)}{0.04\left(\sqrt{2}\right)^{2}-0.024\sqrt{2}+0.0036}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(0.8+\sqrt{2}\right)}{0.04\times 2-0.024\sqrt{2}+0.0036}$$

Multiply 0.04 and 2 to get 0.08.

$$\frac{2\left(0.8+\sqrt{2}\right)}{0.08-0.024\sqrt{2}+0.0036}$$

Add 0.08 and 0.0036 to get 0.0836.

$$\frac{2\left(0.8+\sqrt{2}\right)}{0.0836-0.024\sqrt{2}}$$

Rationalize the denominator of \frac{2\left(0.8+\sqrt{2}\right)}{0.0836-0.024\sqrt{2}} by multiplying numerator and denominator by 0.0836+0.024\sqrt{2}.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{\left(0.0836-0.024\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}$$

Consider \left(0.0836-0.024\sqrt{2}\right)\left(0.0836+0.024\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}.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.0836^{2}-\left(-0.024\sqrt{2}\right)^{2}}$$

Calculate 0.0836 to the power of 2 and get 0.00698896.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00698896-\left(-0.024\sqrt{2}\right)^{2}}$$

Expand \left(-0.024\sqrt{2}\right)^{2}.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00698896-\left(-0.024\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -0.024 to the power of 2 and get 0.000576.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00698896-0.000576\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00698896-0.000576\times 2}$$

Multiply 0.000576 and 2 to get 0.001152.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00698896-0.001152}$$

Subtract 0.001152 from 0.00698896 to get 0.00583696.

$$\frac{2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)}{0.00583696}$$

Divide 2\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right) by 0.00583696 to get \frac{12500000}{36481}\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right).

$$\frac{12500000}{36481}\left(0.8+\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)$$

Use the distributive property to multiply \frac{12500000}{36481} by 0.8+\sqrt{2}.

$$\left(\frac{10000000}{36481}+\frac{12500000}{36481}\sqrt{2}\right)\left(0.0836+0.024\sqrt{2}\right)$$

Use the distributive property to multiply \frac{10000000}{36481}+\frac{12500000}{36481}\sqrt{2} by 0.0836+0.024\sqrt{2} and combine like terms.

$$\frac{836000}{36481}+\frac{1285000}{36481}\sqrt{2}+\frac{300000}{36481}\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{836000}{36481}+\frac{1285000}{36481}\sqrt{2}+\frac{300000}{36481}\times 2$$

Multiply \frac{300000}{36481} and 2 to get \frac{600000}{36481}.

$$\frac{836000}{36481}+\frac{1285000}{36481}\sqrt{2}+\frac{600000}{36481}$$

Add \frac{836000}{36481} and \frac{600000}{36481} to get \frac{1436000}{36481}.

$$\frac{1436000}{36481}+\frac{1285000}{36481}\sqrt{2}$$