Solve frac{(2^{3}*2^{4})^{2}:(2^{4})^8}{(2^5:2^10)^3}

$$\frac { ( 2 ^ { 3 } \cdot 2 ^ { 4 } ) ^ { 2 } : ( 2 ^ { 4 } ) ^ { 8 } } { ( 2 ^ { 5 } : 2 ^ { 10 } ) ^ { 3 } }$$

Evaluate

$$\frac{1}{8}=0.125$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$\frac{\frac{\left(2^{7}\right)^{2}}{\left(2^{4}\right)^{8}}}{\left(\frac{2^{5}}{2^{10}}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{\frac{2^{14}}{\left(2^{4}\right)^{8}}}{\left(\frac{2^{5}}{2^{10}}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 8 to get 32.

$$\frac{\frac{2^{14}}{2^{32}}}{\left(\frac{2^{5}}{2^{10}}\right)^{3}}$$

Rewrite 2^{32} as 2^{14}\times 2^{18}. Cancel out 2^{14} in both numerator and denominator.

$$\frac{\frac{1}{2^{18}}}{\left(\frac{2^{5}}{2^{10}}\right)^{3}}$$

Rewrite 2^{10} as 2^{5}\times 2^{5}. Cancel out 2^{5} in both numerator and denominator.

$$\frac{\frac{1}{2^{18}}}{\left(\frac{1}{2^{5}}\right)^{3}}$$

Calculate 2 to the power of 18 and get 262144.

$$\frac{\frac{1}{262144}}{\left(\frac{1}{2^{5}}\right)^{3}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{\frac{1}{262144}}{\left(\frac{1}{32}\right)^{3}}$$

Calculate \frac{1}{32} to the power of 3 and get \frac{1}{32768}.

$$\frac{\frac{1}{262144}}{\frac{1}{32768}}$$

Divide \frac{1}{262144} by \frac{1}{32768} by multiplying \frac{1}{262144} by the reciprocal of \frac{1}{32768}.

$$\frac{1}{262144}\times 32768$$

Multiply \frac{1}{262144} and 32768 to get \frac{1}{8}.

$$\frac{1}{8}$$

Factor

$$\frac{1}{2 ^ {3}} = 0.125$$