Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)
OPERATIONS OF BASES WITH EXPONENTS
Multiplying Bases With Exponents
When multiplying numbers or variables with a common base, keep the base the same and add the exponents together. Remember… when multiplying factors (with a common base) with exponents, you do not multiply their exponents; this is a frequently made mistake.
Dividing Bases With Exponents
When dividing numbers or variables with a common base, keep the base the same and subtract the exponent of the denominator from the exponent in the numerator. Remember to keep the signs of the exponents. You may subtract a negative exponent, yielding a positive exponent. You may also get a negative number as the exponent, which is fine, but in the final, simplified form of your answer, you shouldn’t leave an exponent negative. If an exponent is negative, move the factor (the base) with that exponent to the opposite part of the fraction and change the sign of the exponent to positive.
Remember, any and every factor has an unwritten exponent of “1”. Also remember that when exponents add or subtract to equal zero, any base to the power zero equals 1. (Review this in: The Unwritten 1, and: Property Crises of Zeros, Ones and Negatives).
Exponents of Exponents (a.k.a. Powers of Powers)
When you take a power to a power, multiply the exponents. Remember that if there are multiple factors, you must distribute the outer exponent to the exponent of each factor in the parentheses, including the coefficient. To distribute the outer exponent to each exponent of factors in the parentheses means you multiply those exponents. Remember, a variable with no exponent shown really is to the power of 1, and must not be forgotten to be multiplied by the outer exponent. This is a frequently made mistake.
Also, when distributing an exponent, do not forget to apply that power to the coefficient if there is one. This is another common mistake, often forgotten by students. This may be because students look for the conspicuous exponents written with the obvious variables, but when coefficients don’t have exponents associated with them, they are just shown with an inconspicuous unwritten power of 1.
you attempt to divide numerators across the top and divide denominators across the bottom (in the way you would do when multiplying fractions), you will notice… it works! However… you are not encouraged to do it that way for one simple reason: it can get very complicated along the way, giving you strange fractions to manage, and many places to make a mistake. For this reason, you are highly encouraged to closely follow the procedure of flipping, then multiplying. It’s easier, and if nothing else, it is much faster.