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(Make up a random partner + their work.)

Question 1
Create two expressions using the following criteria:

Include one expression that contains rational exponents that can be simplified further. Provide the simplified form to your instructor.
Include one expression that contains radicals that can be simplified further. Provide the simplified form to your instructor.

Question 2
Rewrite in simplest radical form
x
5
6
x
1
6
. Show each step of your process.

Question 3
Which of the following expressions are equivalent? Justify your reasoning.

4√x3

1
x−1

10√x5•x4•x2

x
1
3
•x
1
3
•x
1
3

Question 4
Using the technology of your choice, exchange the expressions you created for Question 1 (do not provide the simplified form) and your work for Questions 2 and 3 with a partner. Discuss your findings.

Simplify your partner's expressions for Question 1.
Review and critique your partner's work on Question 2.
Review and critique your partner's work on Question 3.

1 answer

Random Partner Name

Let's create a random partner: Jordan Lee, a high school mathematics teacher with a focus on algebra and calculus.

Question 1

Expression with Rational Exponents:

  1. \( x^{\frac{3}{4}} \cdot x^{\frac{1}{2}} \)

    Simplified form: \[ x^{\frac{3}{4} + \frac{1}{2}} = x^{\frac{3}{4} + \frac{2}{4}} = x^{\frac{5}{4}} \]

Expression with Radicals: 2. \( \sqrt{18} \)

Simplified form: \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]

Question 2

Expression: \[ x^{\frac{5}{6}} \cdot x^{\frac{1}{6}} \]

Step 1: Combine the exponents using the product of powers property: \[ x^{\frac{5}{6} + \frac{1}{6}} \]

Step 2: Add the exponents: \[ x^{\frac{6}{6}} \]

Step 3: Simplify: \[ x^1 = x \]

Question 3

Evaluate the Expressions:

  1. \( 4\sqrt{x^3} \)
  2. \( \frac{1}{x^{-1}} \) = \( x \)
  3. \( 10\sqrt{x^5 \cdot x^4 \cdot x^2} = 10\sqrt{x^{5 + 4 + 2}} = 10\sqrt{x^{11}} = 10x^{\frac{11}{2}} \)
  4. \( x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^1 = x \)

Determine Equivalence:

  • \( 4\sqrt{x^3} \) is not equivalent to \( x \) or \( 10x^{\frac{11}{2}} \).
  • \( \frac{1}{x^{-1}} = x \) is equivalent to the fourth expression \( x \).
  • \( 10\sqrt{x^5 \cdot x^4 \cdot x^2} \) (which simplifies to \( 10x^{\frac{11}{2}} \)) is not equivalent to any of the other expressions.

Justification:

  • The expressions \( \frac{1}{x^{-1}} \) and \( x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \) are equivalent, both simplifying to \( x \).

Question 4

Instructions for Exchange:

  • Share the expressions and work for Question 1 with Jordan.
  • Discuss findings:
    • Ensure to simplify Jordan's expressions for Question 1, if needed.
    • Review and critique Jordan's work on Question 2.
    • Review and critique Jordan's work on Question 3, identifying any equivalences or errors.

Now you have a complete set of responses and instructions for engaging with your partner, Jordan Lee!