Solve frac{{left(sqrt{left(6+2right)3}+2right)}^2}{left(75^2right)^3}

$$\frac{ { \left( \sqrt{ \left( 6+2 \right) 3 } +2 \right) }^{ 2 } }{ { \left(7 { 5 }^{ 2 } \right) }^{ 3 } }$$

Evaluate

$$\frac{8\sqrt{6}}{5359375}+\frac{4}{765625}\approx 0.000008881$$

Solution Steps

Add 6 and 2 to get 8.

$$\frac{\left(\sqrt{8\times 3}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Multiply 8 and 3 to get 24.

$$\frac{\left(\sqrt{24}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{\left(2\sqrt{6}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{6}+2\right)^{2}.

$$\frac{4\left(\sqrt{6}\right)^{2}+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

The square of \sqrt{6} is 6.

$$\frac{4\times 6+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

Multiply 4 and 6 to get 24.

$$\frac{24+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

Add 24 and 4 to get 28.

$$\frac{28+8\sqrt{6}}{\left(7\times 5^{2}\right)^{3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{28+8\sqrt{6}}{\left(7\times 25\right)^{3}}$$

Multiply 7 and 25 to get 175.

$$\frac{28+8\sqrt{6}}{175^{3}}$$

Calculate 175 to the power of 3 and get 5359375.

$$\frac{28+8\sqrt{6}}{5359375}$$

Expand

$$\frac{8 \sqrt{6}}{5359375} + \frac{4}{765625} = 8.881E-06$$

Solution Steps

Add 6 and 2 to get 8.

$$\frac{\left(\sqrt{8\times 3}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Multiply 8 and 3 to get 24.

$$\frac{\left(\sqrt{24}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{\left(2\sqrt{6}+2\right)^{2}}{\left(7\times 5^{2}\right)^{3}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{6}+2\right)^{2}.

$$\frac{4\left(\sqrt{6}\right)^{2}+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

The square of \sqrt{6} is 6.

$$\frac{4\times 6+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

Multiply 4 and 6 to get 24.

$$\frac{24+8\sqrt{6}+4}{\left(7\times 5^{2}\right)^{3}}$$

Add 24 and 4 to get 28.

$$\frac{28+8\sqrt{6}}{\left(7\times 5^{2}\right)^{3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{28+8\sqrt{6}}{\left(7\times 25\right)^{3}}$$

Multiply 7 and 25 to get 175.

$$\frac{28+8\sqrt{6}}{175^{3}}$$

Calculate 175 to the power of 3 and get 5359375.

$$\frac{28+8\sqrt{6}}{5359375}$$