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 575 – 577, do the following problem
Simplify each expression. Write your answers with positive exponents. Assume that all variables represent positive real numbers.
On pages 584 – 585, do the following problem
Write each product as a single radical expression
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.
INSTRUCTOR GUIDANCE EXAMPLE: Week Three Discussion Simplifying Radicals
1. Simplify each expression using the rules of exponents and explain the steps you are taking.
2. Next, write each expression in the equivalent radical form and demonstrate how it can be simplified in that form, if possible.
3. Which form do you think works better for the simplification process and why? #51. (2-4)1/2 The exponent working on an exponent calls for the Power Rule.
2(-4*1/2) The exponents multiply each other. 2-2 -4*1/2 = -2 so the new exponent is -2. 1 22 The negative exponent makes a reciprocal of base number and
exponent. 1 The final simplified answer is ¼. This is the principal root of the 4 square root of 2-4.
#63. 4
1
20
1281
y
x The Power Rule will be used again with the outside exponent
4
1 20
4
1 12
4
1 4
3
y
x multiplying both the inner exponents. 81 = 34
5
33
y
x 4*1/4 = 1, 12*1/4 = 3, and 20*1/4 = 5
All inner exponents were multiples of 4 so no rational exponents are left.
#89. 3
2
27
8
First rewrite each number as a prime to a power.
3
2
3
3
3
2
Use the Power Rule to multiply the inner exponents.
The negative has to be dealt with somewhere so I will put it with the 2 in the numerator.
3
2 3
3
2 3
3
2
3*2/3 = 2 in both numerator and denominator.
9
4
3
2 2
2
The squaring eliminates the negative for the answer.
It turns out that the examples I chose to work out here didn’t use all of the vocabulary words and required one which wasn’t on the list. Students should be sure to use words appropriate to the examples they work on.