Find the value of $(-2)^5 \times (2)^{-5} \times (3)^3$

### Concept & Formula This problem tests the basic laws of indices, specifically dealing with negative bases raised to odd powers, and the product rule for identical bases with different exponents. $$(-a)^{\text{odd}} = -a^{\text{odd}}$$ $$a^m \times a^n = a^{m+n}$$ ### Step-by-Step Solution * **Given:** Expression to evaluate: $(-2)^5 \times (2)^{-5} \times (3)^3$ * **Calculation / Deduction:** 1. First, handle the negative sign in the first term. Since a negative number is raised to an odd power (5), the result will be negative. $$(-2)^5 = - (2^5)$$ 2. Substitute this back into the original expression: $$- (2^5) \times (2)^{-5} \times (3)^3$$ 3. Group the terms with the same base (base 2). Apply the product rule of exponents ($a^m \times a^n = a^{m+n}$): $$- (2^{5 + (-5)}) \times (3)^3$$ $$- (2^0) \times (3)^3$$ 4. Apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is 1 ($a^0 = 1$): $$- (1) \times (3)^3$$ 5. Finally, calculate the cube of 3 ($3 \times 3 \times 3 = 27$) and multiply by -1: $$-1 \times 27 = -27$$ ### Exam Strategy & Shortcut **Mental Math Simplification:** You can solve this mentally in 5 seconds. Recognize that $(-2)^5$ is simply $-32$ and $2^{-5}$ is $\frac{1}{32}$. Multiplying a number by its exact reciprocal yields 1, and preserving the negative sign gives $-1$. So the entire base-2 portion simply cancels out to $-1$. You are left with $-1 \times 3^3 = -27$. ### Common Pitfall A common error is mishandling the negative base. Students sometimes incorrectly assume that $(-2)^5 \times (2)^{-5}$ simplifies to $(-2 \times 2)^{5-5}$ or similar flawed logic, losing track of the crucial negative sign that persists purely because the exponent is an odd number. ### Final Answer **Therefore, the correct answer is $-27$.**