\[ (-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4 - 1 + 6 = 0 \]

["Understanding the Expression: ((-1)^3 - 4(-1)^2 + (-1) + 6 = 0 – A Step-by-Step Breakdown", "Solving algebraic expressions manually is a fundamental skill, especially when working with powers and operations involving negative numbers. One particular expression, [(-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4 - 1 + 6 = 0,] offers an excellent opportunity to review basic exponent rules, arithmetic with negatives, and how to verify solutions efficiently.", "---", "### What Does the Expression Mean?", "The given expression is:\n[(-1)^3 - 4(-1)^2 + (-1) + 6]", "This contains powers, multiplication by constants, and addition or subtraction. To simplify, we begin by evaluating each term using precise exponent and coefficient handling.", "---", "### Step 1: Evaluate Exponents", "Recall that ((-1)^n) behaves differently depending on whether (n) is even or odd:", "- ((-1)^3 = -1) because an odd power keeps the negative sign.\n- ((-1)^2 = 1) because squaring a negative yields a positive.", "So:\n[\n(-1)^3 = -1 \quad \ ext{and} \quad (-1)^2 = 1\n]", "Substitute these values into the expression:\n[\n(-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4(1) + (-1) + 6\n]", "---", "### Step 2: Carry Out Multiplications", "Distribute the (-4) across ((-1)^2 = 1):\n[\n-4 \ imes 1 = -4\n]", "Now replace in the expression:\n[\n-1 - 4 + (-1) + 6\n]", "Note that (-4 + (-1)) simplifies to (-5):\n[\n-1 - 5 + 6\n]", "---", "### Step 3: Perform Addition and Subtraction from Left to Right", "First combine (-1 - 5 = -6):\n[\n-6 + 6 = 0\n]", "So, the entire expression simplifies to zero:\n[\n(-1)^3 - 4(-1)^2 + (-1) + 6 = 0\n]", "---", "### Why Is This Important?", "This exercise demonstrates:", "- Order of operations: Properly evaluating exponents before multiplication and addition.\n- Sign rules: How powers affect the sign (negative to positive only with odd exponents).\n- Verification: Confirming the result by step-by-step simplification ensures accuracy in algebraic manipulation.", "---", "### In Practice: Solving Simple Equations", "The equation ((-1)^3 - 4(-1)^2 + (-1) + 6 = 0) is not just a calculation—it’s a small but complete example of solving for equality using arithmetic. In real math, such simplifications validate whether a value satisfies the equation. Here, since the result is zero, the expression equals zero, confirming correctness.", "---", "### Key Takeaways", "- Always evaluate exponents before performing multiplication or addition.\n- Negative base raised to an odd power remains negative; to even keeps positive.\n- Break complex expressions into simpler steps for clarity.\n- Simplifying ensures your solution is verified and accurate.", "---", "Conclusion", "The algebraic journey from ((-1)^3 - 4(-1)^2 + (-1) + 6) to zero reveals foundational skills every student and learner must master. By following clear rules of exponents and arithmetic, even negative expressions lose their complexity. Whether you’re solving equations, coding, or learning math fundamentals, clearstep reasoning leads to confident results.", "---", "Keywords:\nalgebra simplification, exponent rules negative numbers, solve equation step-by-step, how to simplify (-1)^3, verify algebraic expression, arithmetic with negative exponents, step-by-step math solution, equation validation."]