\( 1-\sqrt {^{2^{-3}+\frac{13}{4}}} \)

To simplify the expression \( 1 - \sqrt{2^{-3} + \frac{13}{4}} \), we first calculate \( 2^{-3} \), which is equal to \( \frac{1}{2^3} = \frac{1}{8} \). Now, we add \( \frac{1}{8} \) to \( \frac{13}{4} \). To add these two fractions, we need a common denominator. The least common multiple of 8 and 4 is 8. Therefore, we convert \( \frac{13}{4} \) to eighths: \[ \frac{13}{4} = \frac{13 \times 2}{4 \times 2} = \frac{26}{8} \] Now we can add: \[ 2^{-3} + \frac{13}{4} = \frac{1}{8} + \frac{26}{8} = \frac{1 + 26}{8} = \frac{27}{8} \] Next, we find the square root: \[ \sqrt{\frac{27}{8}} = \frac{\sqrt{27}}{\sqrt{8}} = \frac{3\sqrt{3}}{2\sqrt{2}} = \frac{3\sqrt{3} \sqrt{2}}{4} = \frac{3\sqrt{6}}{4} \] Finally, we substitute back into the original expression: \[ 1 - \sqrt{2^{-3} + \frac{13}{4}} = 1 - \frac{3\sqrt{6}}{4} \] To express this as a single fraction: \[ 1 = \frac{4}{4} \quad \Rightarrow \quad 1 - \frac{3\sqrt{6}}{4} = \frac{4 - 3\sqrt{6}}{4} \] So, the simplified form of the expression \( 1 - \sqrt{2^{-3} + \frac{13}{4}} \) is: \[ \frac{4 - 3\sqrt{6}}{4} \]