➕ Algebra 1 · Exponents and Polynomials

Zero and Negative Exponents: true or false

Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.

All statements and answers

• x⁰ = 1 for every number x.

  ✗ False. It is true for every nonzero x. The expression 0⁰ is left undefined in this course.

• A negative exponent always makes the answer a negative number.

  ✗ False. No. A negative exponent means take a reciprocal. For example, 20⁻¹ = 1/20, which is positive.

• −12⁰ = 1.

  ✗ False. The exponent touches only the 12, so −12⁰ = −(12⁰) = −1. With parentheses, (−12)⁰ = 1.

• (−4)⁻² = 1/16.

  ✓ True. (−4)⁻² = 1/(−4)² = 1/16. Squaring a negative number gives a positive number.

• 1/x⁸ = x⁻⁸ when x ≠ 0.

  ✓ True. A negative exponent means one over the power with the positive exponent, so x⁻⁸ = 1/x⁸.

• 2y⁻³ = 2/y³ when y ≠ 0.

  ✓ True. Only y has the −3, so only y moves to the bottom. The 2 stays on top.

• 4x⁰ = 1 when x ≠ 0.

  ✗ False. Only x is raised to the 0 power, so 4x⁰ = 4 · 1 = 4. It is (4x)⁰ that equals 1.

• x⁻² · x⁻³ = x⁻⁵ when x ≠ 0.

  ✓ True. Product rule: add the exponents, −2 + (−3) = −5. That is the same as 1/x⁵.

• x⁻² + x⁻³ = x⁻⁵ when x ≠ 0.

  ✗ False. The product rule is for multiplying. A sum cannot be combined by adding exponents. You can rewrite it as 1/x² + 1/x³ (or (x + 1)/x³), but not as x⁻⁵.

• (3/8)⁻¹ = 8/3.

  ✓ True. A negative exponent on a fraction flips it: (3/8)⁻¹ = (8/3)¹ = 8/3.

• 0⁻² equals 0.

  ✗ False. 0⁻² = 1/0² = 1/0, which is undefined, because you cannot divide by 0.

• Since x⁸ ÷ x⁸ = 1 and the quotient rule would give x⁸⁻⁸ = x⁰, the quotient rule keeps working only if x⁰ = 1 (for x ≠ 0).

  ✓ True. This is the reasoning behind the zero exponent rule: we choose x⁰ = 1 so that the quotient rule still works when the two exponents are equal.