§7.4—Working with Rational Exponents
December 10, 2012
Snow Day “Lesson” – December 10, 2012
Lesson objective: Evaluate any numerical expression involving exponents which are rational
numbers (i.e. which may be fractions or decimals).
[Ministry expectation B1.2]
Refer to “Investigate the Math” in your textbook on page 410. Mathematical models help us to
determine conditions which exist in the intervals between actual observations. This is an example
of a case where we use a mathematical pattern to estimate a population level at a point in time
between two documented values.
The population of Dawson City is shown in a table, and graphed. The independent variable, x, is
the number of quarters1 (i.e. quarter-years) since April 1, 1896. Looking at the table, we see that
during the time span covered by the data, the population triples every quarter-year. We can
therefore express the population at a given x value using the formula P = 3x, as shown by the last
column of the table.
The table gives us the population at the beginning of April, July, October, and January (1897).
But suppose we wanted to estimate the population at other dates within this timeframe. (That is,
we wanted to use interpolation to determine additional population values – remember that term
from grade 9?) We could do that using the graph, but a more accurate estimate could be
obtained by calculation using the formula P = 3x.
In the example, we are asked about the population in mid-May, 1896. That is 1½ months after
April 1. Since on July 1 (3 months after April 1) x = 1, in mid-May, x will be equal to ½ (or 0.5).
If we wanted to use the formula to calculate the population, we would have to evaluate 3. .
But what does it mean to have an exponent of 0.5 (or ½)? We’ll use the exponent law for
multiplication to find out.
We know that when we multiply powers having the same base, we add the exponents, so for
example, 32 • 33 = 35. Using this rule, it is clear that
3. · 3. 3 (or just 3)
But we also know that
√3 · √3 3
Comparing these two equations, we see that √3 must be the same as 3. .
The above discussion should enable you to answer questions A - C on page 411. Answer these
questions in your notebook, then work through the remaining questions D - K.
1
Note: Corporations often report their financial results, and predict future performance, using quarter-year intervals.
MCF3M—S. Inrig
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§7.4—Working with Rational Exponents
December 10, 2012
You should have concluded that a power having an exponent of
the base, that is,
is equivalent to the nth root of
√
Look at Example 1 on page 411.. You must use a calculator to get an accurate answer, so you
will need to determine how to take cube roots on your particular calculator. A couple of
commonly used keys are shown below.
TI-30X
Sharp
With these calculators, you could enter 1 / 3 as the exponent and use the ^ or yx key to calculate
the cube root. Using the “2nd” or “2ndF” key, you could enter 3 rather than 1 / 3 to take a cube
root. Other calculator models may use different methods, so check your calculator manual.
In Example 2 (page 412), you must change the decimal exponent into a fraction if the base is
negative—most
most calculators will give an error if you attempt to calculate 32... Note that
when the base of a power is negative, the value will not be a real number if the numerator of the
exponent
nt fraction is odd and the denominator is even.
Look at Example 3 on page 413.. “George’s Solution” would be better done as:
27 27 √27
3 Finally, look at Example 4 on page 4414. Use exponent laws for this type of question, not a
calculator.
The “In Summary” section on page 4415 recaps the lesson.
You should now be ready to do the homework assignment as indicated on the web page:
P. 415: #1-3,
3, 4bdf, 5, 6def, 9, 10df, 11d, 15, 17
Iff you have difficulty with the assignment, be sure to contact me either by phone (613-738-5322)
(613
or by email ([email protected]).
MCF3M—S. Inrig
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