Question

Part 1: Explain, in complete sentences, how multiplying and dividing factors with rational exponents is similar to multiplying and dividing factors with integer exponents. Part 2: Give one example of multiplying factors with rational exponents. Part 3: Give one example of multiplying factors with integer exponents.

Answer 1

The basic rules are the same for integer and rational exponents. For example if you multiply like bases you still would add the exponents, if you divide you would subtract the exponents. It is the interpretation of integer vs, rational exponents that differs. An integer exponent represents a repetitive multiplication of the base. Consider,\[a \times a \times a = a^3\]The exponent of 3 indicates that there were three factors of a. Rational exponents represent roots. Consider the following,\[\sqrt{a}=a ^{1/2} \]\[\sqrt[3]{a} = a ^{1/3}\]Of course the two can be combined. Suppose we squared a cube root:\[\left( \sqrt[3]{a} \right)^2=a ^{2/3}\]
Here is a similar example with numbers:\[8^{2/3}=\sqrt[3]{8}^2=2^{2}=4\]

Answer 2

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