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In this discussion, you will simplify and compare equivalent expressions written both in radical form and with rational (fractional) exponents. Read the following instructions in order and view the example (available for download in your online classroom) to complete this discussion. Please complete the following problems according to your assigned number. (Instructors will assign each student their number.)
|
On pages 5 75 – 577, do the following problem |
On pages 584 – 585, do the following problem |
5 |
10 2 |
Simplify each expression using the rules of exponents and examine the steps you are taking.
Incorporate the following five math vocabulary words into your discussion. Use boldfont to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing the thought behind your math work.
oPrincipal root
oProduct rule
oQuotient rule
oReciprocal
onth root
Refer to
Inserting Math Symbols
for guidance with formatting. Be aware with regards to the square root symbol, you will notice that it only shows the front part of a radical and not the top bar. Thus, it is impossible to tell how much of an expression is included in the radical itself unless you use parenthesis. For example, if we have √12 + 9 it is not enough for us to know if the 9 is under the radical with the 12 or not. Therefore, we must specify whether we mean it to say √(12) + 9 or √(12 + 9), as there is a big difference between the two. This distinction is important in your notation.
Another solution is to type the letters “sqrt” in place of the radical and use parenthesis to indicate how much is included in the radical as described in the second method above. The example above would appear as either “sqrt(12) + 9” or “sqrt(12 + 9)” depending on what we needed it to say.
Your initial post should be at least 250 words in length. Support your claims with examples from required material(s) and/or other scholarly resources, and properly cite any references
I have attached the pages from the book that are required.
When simplifying expressions involving rational exponents and variables, we must be careful to write
equivalent expressions. For example, in the equation
Helpful Hint
We usually think of squaring and taking a square root as inverse operations, which they are as long as we
stick to positive numbers. We can square 3 to get 9, and then find the square root of 9 to get 3—what we
started with. We don’t get back to where we began if we start with −3.
it looks as if we are correctly applying the power of a power rule. However, this statement is false if x is
negative because the 1/2 power on the left-hand side indicates the positive square root of x2. For
example, if x = −3, we get
which is not equal to −3. To write a simpler equivalent expression for (x2)1/2, we use absolute value as
follows.
Square Root of x2
For any real number x,
Note that both
and
are identities. They are true whether x is positive, negative, or zero.
It is also necessary to use absolute value when writing identities for other even roots of expressions
involving variables.
Page 574
EXAMPLE 8
Using absolute value symbols with roots
Simplify each expression. Assume the variables represent any real numbers and use absolute value
symbols as necessary.
a)
b)
Solution
a) Apply the power of a product rule to get the equation (x8y4) = x2y. The
left-hand side is nonnegative for any choices of x and y, but the righthand side is negative when y is negative. So for any real values of x
and y we have
Note that the absolute value symbols could also be placed around
the entire expression:
b) Using the power of a quotient rule, we get
This equation is valid for every real number x, so no absolute value
signs are used.
Now do Exercises 55–64
Because there are no real even roots of negative numbers, the expressions
are not real numbers if the variables have negative values. To simplify matters, we sometimes assume
the variables represent only positive numbers when we are working with expressions involving variables
with rational exponents. That way we do not have to be concerned with undefined expressions and
absolute value.
EXAMPLE 9
Expressions involving variables with rational exponents
Use the rules of exponents to simplify the following. Write your answers with positive exponents. Assume
all variables represent positive real numbers.
a)
b)
c)
d)
Solution
a)
b)
c)
d) Because this expression is a negative power of a quotient, we can
first find the reciprocal of the quotient and then apply the power of a
power rule:
Page 576
Page 577
a)Find D for the box shown in the accompanying figure.
b)Find D if L = W = H = 1 inch.
Page 578
Page 583
a)
b)
a)
b)
Page 584
Page 585
Page 586
