Why this matters for FAST: Exponent rules are essential for simplifying algebraic expressions and understanding scientific notation. FAST tests require students to apply these rules correctly and recognize equivalent expressions.
Students think x² times x³ = x⁶ (multiply) instead of x⁵ (add exponents).
"When multiplying same bases, ADD the exponents. Why? x² times x³ = (x times x) times (x times x times x) = 5 x's total = x⁵. You're counting HOW MANY x's you have!"
Students think x⁰ = 0 instead of x⁰ = 1.
"Any non-zero number to the 0 power equals 1, NOT 0! Pattern: 2³=8, 2²=4, 2¹=2, 2⁰=1 (dividing by 2 each time). Also: x³/x³ = x^(3-3) = x⁰, but x³/x³ = 1!"
Students think x⁻² is a negative number instead of 1/x².
"Negative exponent means RECIPROCAL, not negative! x⁻² = 1/x². Think: The negative tells you to flip it to the denominator. 2⁻³ = 1/2³ = 1/8 (positive!)"
Example: x⁴ times x³ = x^(4+3) = x⁷
Example: 2³ times 2⁵ = 2⁸ = 256
Example: x⁷ / x³ = x^(7-3) = x⁴
Example: 5⁶ / 5² = 5⁴ = 625
Example: (x³)⁴ = x^(3 times 4) = x¹²
Example: (2²)³ = 2⁶ = 64
Zero Exponent: x⁰ = 1 (for x ≠ 0)
Negative Exponent: x⁻ⁿ = 1/xⁿ
Examples: 5⁰ = 1 | 3⁻² = 1/3² = 1/9 | x⁻⁴ = 1/x⁴
Scientific notation uses powers of 10: 3.2 times 10⁴ = 32,000
Very small numbers: 5 times 10⁻³ = 0.005
"Simplify: x⁵ times x³"
Solution: x⁵⁺³ = x⁸