Students have already had an introduction to the quotient of powers in AE 11 . In that lesson, students learned the first part of the rule of exponents for division:
In this lesson, students are presented with the second part of the rule:
Warm-Ups.
Problem 3 will come up again in a later unit as a special product. Problem 5 is a complex fraction. The answer shows that it is treated as a division, with the end result being the reciprocal of 2/5. Students will later learn that 1 divided by any number is the reciprocal of that number.
Rule of Exponents for Division. I write on the board.
“We learned that the expression I just wrote is the same as this one.”
“We can divide two b’s on the top and two b’s on the bottom and we get b⁵. I want to write the above in a different way.”
“I can rewrite this by making this the product of two fractions.”
“What’s b²/b²?”
Hearing “one”, we are left with 1∙ b⁵/1 which is just b⁵/1 .
“I could write this as b⁵ but I have my reasons for writing it as I did, which I shall now reveal.”
I write on the board:
“What do you think the answer will be?”
Generally students will follow the pattern and say it is 1/ b⁵. Some will say that it is b⁷/b² upside down—which it is. Therefore the answer will be upside down.
“Which brings us to the second part of the rule of exponents for division.”
“You can think of it this way. For b⁷/b², we have more b’s on top than we have on the bottom, so we subtract ‘downward’. That is b to the 7-2. But when we have b²/b⁷ we have more b’s on the bottom. So we end up with 1 in the numerator; in the denominator we subtract ‘upward’. In this case, b to the power of 7-2, which is b to the power of 5, is in the denominator.”
Examples:
Students are reminded to break out expressions into fractions that contain numbers and same variable bases as shown above. After a while they will be able to do that in their heads, but in displaying answers I show the process to remind them.
In this case, there is no z variable in the denominator so we write it as z/1—it just stays as is in the numerator. Also, whether there is a negative in the numerator or denominator, the entire fraction becomes negative.
Homework. I work with the students on the homework problems as they start in on it. The homework should be a mix of problems in which the exponents in the denominator are greater than those in the numerator for the same base, and vice versa. Also there should be problems about products of powers and powers of products which were covered in the previous lessons.