Question 1

Question

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.

Answer 1

Sure, I can help you with that. Here are the answers to your questions:

Question 1

Rational Exponent Expression:

1. Expression: \( x^{\frac{3}{2}} \)
   • Simplified Form: \( \sqrt{x^3} \)

Radical Expression: 2. Expression: \( \sqrt{50} \)

• Simplified Form: \( 5\sqrt{2} \) (since \( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \))

Question 2

To rewrite \( x^{\frac{5}{6}} \cdot x^{\frac{1}{6}} \) in simplest radical form, follow these steps:

1. Combine the exponents (since the bases are the same): \[ x^{\frac{5}{6}} \cdot x^{\frac{1}{6}} = x^{\frac{5}{6} + \frac{1}{6}} = x^{\frac{6}{6}} = x^1 \]

2. Convert to radical form:

   • Since \( x^1 \) is simply \( x \), in radical form it remains: \[ x = \sqrt[6]{x^6} \equiv \sqrt[6]{x^6} \]

   This radical form is only necessary if you were initially asked to keep it in radical terms. The simplest form is \( x \).

Question 3

Let's evaluate which of the given expressions are equivalent:

1. Expression 1: \( 4\sqrt{x^3} \) can be rewritten as: \[ 4x^{3/2} \]

2. Expression 2: \( \frac{1}{x^{-1}} \) simplifies to: \[ x^1 \text{ (since } \frac{1}{x^{-1}} = x^{1}\text{)} \]

3. Expression 3: \( 10\sqrt{x^5} \cdot x^4 \cdot x^2 \)

   • Simplifying: \[ = 10x^{5/2} \cdot x^4 \cdot x^2 = 10x^{5/2 + 4 + 2} = 10x^{\frac{5}{2} + \frac{8}{2} + \frac{4}{2}} = 10x^{\frac{17}{2}} \]

4. Expression 4: \( x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \) simplifies to: \[ x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^{1} \]

• Now, comparing the simplified forms:
  • \( 4x^{3/2} \) is not equivalent to \( x^1 \).
  • \( x^1 \) is equivalent to itself.
  • \( 10x^{\frac{17}{2}} \) is not equivalent to \( x^{1} \).
  • \( x^1 \) from Expression 4 matches the second expression.

Summary of Equivalents:

• The only equivalent expressions are \( \frac{1}{x^{-1}} \) and \( x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \) (or, simply \( x^1 \)).

Question 4:

Discuss your findings with your partner by reviewing and critiquing their expressions for Question 1, and the steps you followed in Questions 2 and 3. Ensure to offer constructive feedback and double-check their simplifications and reasoning for similarities in expressions!