Overview
Repeated multiplication explains exponent rules: multiplying equal bases concatenates their factors, whereas a power of a power repeats the full group. Negative exponents express reciprocals and require nonzero bases. Scientific notation separates a coefficient from a power of 10 so place value can be tracked.
Consider this example: For x ≠ 0, simplify x⁵/x². Cancel two factors of x from numerator and denominator, leaving x³. Subtract exponents 5 − 2 = 3; adding them would apply a multiplication rule to division.
The method also handles a different case: Simplify (−2a³b)². Squaring applies to every factor: (−2)² = 4, (a³)² = a⁶, and b² remains. The result is 4a⁶b²; keeping a negative coefficient fails to square its sign.
For a further application, consider: For positive x, simplify x¹ᐟ² · x³ᐟ². The positive-base condition permits the familiar exponent rule. Add 1/2 + 3/2 = 2, giving x². Multiplying the exponents would incorrectly yield 3/4.
Multiplying exponents when multiplying equal bases confuses two different rules. A negative exponent changes where the factor appears; it does not make the value negative. Identify whether you are multiplying powers, raising a power, or taking a reciprocal. For a new expression, compare with a small positive value of the variable to detect a sign or exponent error.
A negative exponent expresses a reciprocal, not a negative value. To evaluate 2⁻³ + 2⁻², first write 1/8 + 1/4. Give the fractions a common denominator: 1/8 + 2/8 = 3/8. Adding the exponents here would be wrong because the outer operation is addition, not multiplication. In scientific notation, divide the coefficients and powers separately. For (6 × 10⁵)/(3 × 10²), 6/3 = 2 and 10⁵/10² = 10³, so the result is 2 × 10³ = 2000. Multiplying 2000 by the original denominator 300 restores 600000, the numerator. This check catches both a reversed exponent subtraction and a misplaced decimal point.