Solve frac{sqrt{4^{-4}+4^{4}+4^{4}+4^{4}}}{sqrt{4^5+4^3+4^5+4^5}}=

$$\frac { \sqrt { 4 ^ { - 4 } + 4 ^ { 4 } + 4 ^ { 4 } + 4 ^ { 4 } } } { \sqrt { 4 ^ { 5 } + 4 ^ { 3 } + 4 ^ { 5 } + 4 ^ { 5 } } } =$$

Evaluate

$$\frac{\sqrt{196609}}{896}\approx 0.494872918$$

Solution Steps

Calculate 4 to the power of -4 and get \frac{1}{256}.

$$\frac{\sqrt{\frac{1}{256}+4^{4}+4^{4}+4^{4}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Calculate 4 to the power of 4 and get 256.

$$\frac{\sqrt{\frac{1}{256}+256+4^{4}+4^{4}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Add \frac{1}{256} and 256 to get \frac{65537}{256}.

$$\frac{\sqrt{\frac{65537}{256}+4^{4}+4^{4}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Calculate 4 to the power of 4 and get 256.

$$\frac{\sqrt{\frac{65537}{256}+256+4^{4}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Add \frac{65537}{256} and 256 to get \frac{131073}{256}.

$$\frac{\sqrt{\frac{131073}{256}+4^{4}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Calculate 4 to the power of 4 and get 256.

$$\frac{\sqrt{\frac{131073}{256}+256}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Add \frac{131073}{256} and 256 to get \frac{196609}{256}.

$$\frac{\sqrt{\frac{196609}{256}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Rewrite the square root of the division \sqrt{\frac{196609}{256}} as the division of square roots \frac{\sqrt{196609}}{\sqrt{256}}.

$$\frac{\frac{\sqrt{196609}}{\sqrt{256}}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Calculate the square root of 256 and get 16.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{4^{5}+4^{3}+4^{5}+4^{5}}}$$

Calculate 4 to the power of 5 and get 1024.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{1024+4^{3}+4^{5}+4^{5}}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{1024+64+4^{5}+4^{5}}}$$

Add 1024 and 64 to get 1088.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{1088+4^{5}+4^{5}}}$$

Calculate 4 to the power of 5 and get 1024.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{1088+1024+4^{5}}}$$

Add 1088 and 1024 to get 2112.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{2112+4^{5}}}$$

Calculate 4 to the power of 5 and get 1024.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{2112+1024}}$$

Add 2112 and 1024 to get 3136.

$$\frac{\frac{\sqrt{196609}}{16}}{\sqrt{3136}}$$

Calculate the square root of 3136 and get 56.

$$\frac{\frac{\sqrt{196609}}{16}}{56}$$

Express \frac{\frac{\sqrt{196609}}{16}}{56} as a single fraction.

$$\frac{\sqrt{196609}}{16\times 56}$$

Multiply 16 and 56 to get 896.

$$\frac{\sqrt{196609}}{896}$$