Question: User: [\frac{2^{-4} + 3^{-2}}{2^{-1} - 3^{-1}} : (0,5)^3 - 3^{-1}]^{1/3}
1. **State the problem:** Simplify the expression $$\left[\frac{2^{-4} + 3^{-2}}{2^{-1} - 3^{-1}} : (0.5)^3 - 3^{-1}\right]^{\frac{1}{3}}$$.
2. **Rewrite the expression clearly:** Note that the colon ":" means division, so the expression inside the brackets is
$$\frac{\frac{2^{-4} + 3^{-2}}{2^{-1} - 3^{-1}}}{(0.5)^3 - 3^{-1}}$$
3. **Calculate each component:**
- Calculate powers:
$$2^{-4} = \frac{1}{2^4} = \frac{1}{16}$$
$$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$
$$2^{-1} = \frac{1}{2}$$
$$3^{-1} = \frac{1}{3}$$
$$(0.5)^3 = (\frac{1}{2})^3 = \frac{1}{8}$$
4. **Calculate numerator of the big fraction:**
$$2^{-4} + 3^{-2} = \frac{1}{16} + \frac{1}{9} = \frac{9}{144} + \frac{16}{144} = \frac{25}{144}$$
5. **Calculate denominator of the big fraction's numerator:**
$$2^{-1} - 3^{-1} = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$$
6. **Divide numerator by denominator:**
$$\frac{\frac{25}{144}}{\frac{1}{6}} = \frac{25}{144} \times \frac{6}{1} = \frac{25 \times 6}{144} = \frac{150}{144}$$
7. **Simplify fraction:**
$$\frac{150}{144} = \frac{\cancel{6}25}{\cancel{6}24} = \frac{25}{24}$$
8. **Calculate denominator of the big fraction:**
$$(0.5)^3 - 3^{-1} = \frac{1}{8} - \frac{1}{3} = \frac{3}{24} - \frac{8}{24} = -\frac{5}{24}$$
9. **Divide the two parts:**
$$\frac{\frac{25}{24}}{-\frac{5}{24}} = \frac{25}{24} \times \frac{24}{-5} = \frac{25 \times \cancel{24}}{\cancel{24} \times (-5)} = -\frac{25}{5} = -5$$
10. **Apply the outer exponent:**
$$\left(-5\right)^{\frac{1}{3}} = -\sqrt[3]{5}$$
**Final answer:** $$-\sqrt[3]{5}$$