Exponents and Division – the Quotient of Powers Property – Practice Problems

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The Quotient Property (Quotient of Powers Property, Quotient of Powers Rule) states that we can divide two powers with the same base by simply subtracting their exponents and retaining their common base.  For example, dividing the algebraic expression y^18 by y^12 will give us a result equal to y^(18-12), or y^6, where y can stand for any numerical value except 0; remember that the Quotient Property is only applicable if the expressions share a common base and that base must not be equal to zero. The base cannot be zero because it is not defined by the properties of arithmetic: if 12 could be divided by 0, for instance, then this would mean that the equation 0 times y = 12 would have a solution… which is not possible!

The Quotient Property works because if we do it the “long” way by expanding each expression, y^18 = y•y•y•y•y•y•y•y•y•y•y•y•y•y•y•y•y•y and y^12 = y•y•y•y•y•y•y•y•y•y•y•y, and then divide, we can see that the 12 y’s in the denominator will cancel out with 12 of the y’s in the numerator, leaving only 6 "y"s remaining in the numerator and no y’s (or the number 1) in the denominator.

We can now calculate, for instance, 2^25 / 2^15 = 2^(25-15) = 2^10, or 1024, and that 0^101 / 0^83 just does not make any sense.

The Quotient Property is very useful in simplifying complicated expressions and makes manipulation of large quantities less tedious and less time-consuming.

Know and apply the properties of integer exponents to generate equivalent numerical expressions.
CCSS.MATH.CONTENT.8.EE.A.1

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Exercises in this Practice Problem
Explain how to divide $2^5$ by $2^3$.
Determine the number of hard drives AA-RNO needs to store an entire copy of the internet.
Decide which division the quotient of powers property can be used.
Examine the growth of the internet data volume between 3008 and 8008.
Describe the quotient of powers property.
Carry out the following divisions.