Solve (14+2)^2+8/(2)^-2/4sqrt{3^3+2/3+22^+4}

$$\frac { \frac { ( 14 + 2 ) ^ { 2 } + 8 } { ( 2 ) ^ { - 2 } } } { 4 \frac { \sqrt { 3 ^ { 3 } } + 2 } { 3 + 22 ^ { + 4 } } }$$

Evaluate

$$\frac{61844376\left(3\sqrt{3}-2\right)}{23}\approx 8594089.225355648$$

Solution Steps

Add 14 and 2 to get 16.

$$\frac{\frac{16^{2}+8}{2^{-2}}}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Calculate 16 to the power of 2 and get 256.

$$\frac{\frac{256+8}{2^{-2}}}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Add 256 and 8 to get 264.

$$\frac{\frac{264}{2^{-2}}}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{264}{\frac{1}{4}}}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Divide 264 by \frac{1}{4} by multiplying 264 by the reciprocal of \frac{1}{4}.

$$\frac{264\times 4}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Multiply 264 and 4 to get 1056.

$$\frac{1056}{4\times \frac{\sqrt{3^{3}}+2}{3+22^{4}}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{1056}{4\times \frac{\sqrt{27}+2}{3+22^{4}}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{1056}{4\times \frac{3\sqrt{3}+2}{3+22^{4}}}$$

Calculate 22 to the power of 4 and get 234256.

$$\frac{1056}{4\times \frac{3\sqrt{3}+2}{3+234256}}$$

Add 3 and 234256 to get 234259.

$$\frac{1056}{4\times \frac{3\sqrt{3}+2}{234259}}$$

Express 4\times \frac{3\sqrt{3}+2}{234259} as a single fraction.

$$\frac{1056}{\frac{4\left(3\sqrt{3}+2\right)}{234259}}$$

Divide 1056 by \frac{4\left(3\sqrt{3}+2\right)}{234259} by multiplying 1056 by the reciprocal of \frac{4\left(3\sqrt{3}+2\right)}{234259}.

$$\frac{1056\times 234259}{4\left(3\sqrt{3}+2\right)}$$

Cancel out 4 in both numerator and denominator.

$$\frac{264\times 234259}{3\sqrt{3}+2}$$

Rationalize the denominator of \frac{264\times 234259}{3\sqrt{3}+2} by multiplying numerator and denominator by 3\sqrt{3}-2.

$$\frac{264\times 234259\left(3\sqrt{3}-2\right)}{\left(3\sqrt{3}+2\right)\left(3\sqrt{3}-2\right)}$$

Consider \left(3\sqrt{3}+2\right)\left(3\sqrt{3}-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}.

$$\frac{264\times 234259\left(3\sqrt{3}-2\right)}{\left(3\sqrt{3}\right)^{2}-2^{2}}$$

Multiply 264 and 234259 to get 61844376.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{\left(3\sqrt{3}\right)^{2}-2^{2}}$$

Expand \left(3\sqrt{3}\right)^{2}.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{3^{2}\left(\sqrt{3}\right)^{2}-2^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{9\left(\sqrt{3}\right)^{2}-2^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{9\times 3-2^{2}}$$

Multiply 9 and 3 to get 27.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{27-2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{27-4}$$

Subtract 4 from 27 to get 23.

$$\frac{61844376\left(3\sqrt{3}-2\right)}{23}$$

Use the distributive property to multiply 61844376 by 3\sqrt{3}-2.

$$\frac{185533128\sqrt{3}-123688752}{23}$$

Factor

$$\frac{61844376 {(3 \sqrt{3} - 2)}}{23} = 8594089.225355648$$