Because a negative exponent turns the base into its reciprocal, a simple method for dividing negative exponents is letting the negative cancel and multiplying by the positive exponential expression. Regents Prep notes that when multiple exponents have the same base, the terms can be easily divided or multiplied because the exponents themselves can be added or subtracted.
The positive exponent k states that k amount of the bases are multiplied together. For example, 2^3 = 2*2*2 = 8.
The exponent -k states that 1 is divided by the base k times. For example, 3^(-1) = 1/3 and 3^(-4) = 1/(3*3*3*33) = 1/81. A base to a negative exponent is equal to one divided by that base to the absolute value of the exponent, such that 3^(-4) = 1/(3^4).
Math Is Fun explains that to multiply by negative exponent, one can divide by the positive exponent. For example, (4^3) * (4^(-2)) = (4^3) * 1/(4^2) = (4^3)/(4^2).
To divide by a negative exponent, one treats it like a positive exponent being multiplied. For example (4^3)/(4^(-2)) = (4^3) * 1/(4^(-2)) = (4^3) * 1/(1/(4^2)) because a negative exponent is the reciprocal of the positive, it equals (4^3) * (4^2).
If the exponential expressions have the same base, the exponents can be added when the bases are multiplied and subtracted when the bases are divided. When dividing a negative exponent, one is subtracting a negative number, which behaves like an added positive number.
For example, (4^3) * (4^2) = 4^5. Similarly, (4^3) / (4^(-2)) = 4^5.