Solve [(1/2)^{5}*(1/2)^{2}:(1/2)^{2}]^2:[(1/2)^4*(frac{1}{2})^6:(frac{1}{2})^3]:frac{1}{8}

$$[ ( \frac { 1 } { 2 } ) ^ { 5 } \cdot ( \frac { 1 } { 2 } ) ^ { 2 } : ( \frac { 1 } { 2 } ) ^ { 2 } ] ^ { 2 } : [ ( \frac { 1 } { 2 } ) ^ { 4 } \cdot ( \frac { 1 } { 2 } ) ^ { 6 } : ( \frac { 1 } { 2 } ) ^ { 3 } ] : \frac { 1 } { 8 }$$

Evaluate

$$1$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$\frac{\frac{\left(\frac{\left(\frac{1}{2}\right)^{7}}{\left(\frac{1}{2}\right)^{2}}\right)^{2}}{\frac{\left(\frac{1}{2}\right)^{4}\times \left(\frac{1}{2}\right)^{6}}{\left(\frac{1}{2}\right)^{3}}}}{\frac{1}{8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 7 to get 5.

$$\frac{\frac{\left(\left(\frac{1}{2}\right)^{5}\right)^{2}}{\frac{\left(\frac{1}{2}\right)^{4}\times \left(\frac{1}{2}\right)^{6}}{\left(\frac{1}{2}\right)^{3}}}}{\frac{1}{8}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{\frac{\left(\frac{1}{2}\right)^{10}}{\frac{\left(\frac{1}{2}\right)^{4}\times \left(\frac{1}{2}\right)^{6}}{\left(\frac{1}{2}\right)^{3}}}}{\frac{1}{8}}$$

To multiply powers of the same base, add their exponents. Add 4 and 6 to get 10.

$$\frac{\frac{\left(\frac{1}{2}\right)^{10}}{\frac{\left(\frac{1}{2}\right)^{10}}{\left(\frac{1}{2}\right)^{3}}}}{\frac{1}{8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 10 to get 7.

$$\frac{\frac{\left(\frac{1}{2}\right)^{10}}{\left(\frac{1}{2}\right)^{7}}}{\frac{1}{8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 7 from 10 to get 3.

$$\frac{\left(\frac{1}{2}\right)^{3}}{\frac{1}{8}}$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

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

Divide \frac{1}{8} by \frac{1}{8} to get 1.

$$1$$

Factor

$$1$$