1. **State the problem:** Evaluate the expression $$\frac{-2 \times (3 - 4) + 5 \times (-1) + 2 \times 3 \times 4 - 17}{(\sqrt{2})^4 \times 3^2 - 2^3 \times 2^{-1} + 64^{1/3}}$$ and write the answer in the form $\frac{1}{a}$, stating the integer $a$. 2. **Simplify the numerator:** Calculate each term: - $3 - 4 = -1$ - $-2 \times (-1) = 2$ - $5 \times (-1) = -5$ - $2 \times 3 \times 4 = 24$ Now sum all terms: $$2 + (-5) + 24 - 17 = 2 - 5 + 24 - 17$$ Calculate stepwise: $$2 - 5 = -3$$ $$-3 + 24 = 21$$ $$21 - 17 = 4$$ So, numerator = 4. 3. **Simplify the denominator:** Calculate each part: - $(\sqrt{2})^4 = (2^{1/2})^4 = 2^{(1/2) \times 4} = 2^2 = 4$ - $3^2 = 9$ - $2^3 = 8$ - $2^{-1} = \frac{1}{2}$ - $64^{1/3} = (4^3)^{1/3} = 4$ Now substitute: $$4 \times 9 - 8 \times \frac{1}{2} + 4$$ Calculate each term: $$4 \times 9 = 36$$ $$8 \times \frac{1}{2} = 4$$ So denominator: $$36 - 4 + 4 = 36$$ 4. **Form the fraction:** $$\frac{4}{36}$$ Simplify by dividing numerator and denominator by 4: $$\frac{\cancel{4}^1}{\cancel{36}^9} = \frac{1}{9}$$ 5. **Final answer:** The expression equals $\frac{1}{9}$, so $a = 9$.