Function of the Day:

Question

Given these equations, arrange a, b, c, d, e, f, and g from least to greatest.
Explain your reasoning.

What do you know?

1.

2.

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10.

Vocabulary Terms - Define/Illustrate

Exponent
base
Standard form vs. Exponent form

5 Problems - SHOW ALL WORK!

What does x4 mean?

What operation are you doing?

What does 3-4mean?

Is your answer negative?
What is -25 and how does it compare to (-2)5?

Why are they different?



What is -(-3)3 ?
Write it out in long hand to help you see the correct answer.
What do you think of when you hear the word “exponents”?
What operation?
How is it different?

Example: What is the difference between 3(2) and 3^2 ?

What do I need help with?

Answer 1

Function of the Day: Arranging Values

To arrange the values a, b, c, d, e, f, and g from least to greatest, I'll need specific equations or values for each variable. However, as they are not provided, let's analyze the type of reasoning generally used to order numbers:

1. Identify Values: Obtain the numerical values or results of expressions represented by a, b, c, d, e, f, and g.
2. Compare: Once the values are known, compare them pairwise to determine which is smaller or larger.
3. Sort: Place the values in ascending order based on their comparisons.

Vocabulary Terms

1. Exponent: An exponent is a number that shows how many times a base is multiplied by itself. For example, in \( 3^2 \), the number 2 is the exponent.

2. Base: The base is the number that is raised to a power by the exponent. In \( 3^2 \), 3 is the base.

3. Standard Form vs. Exponent Form:

   • Standard Form: Refers to writing numbers in their usual decimal format, such as 9 instead of \( 3^2 \).
   • Exponent Form: Represents numbers using exponents, such as \( 3^2 \), which means \( 3 \times 3 = 9 \).

Problems

1. What does \( x^4 \) mean?
   \( x^4 = x \times x \times x \times x \). It means the variable \( x \) is multiplied by itself four times.

2. What operation are you doing?
   You are performing multiplication.

3. What does \( 3^{-4} \) mean?
   \( 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \). It indicates the reciprocal of \( 3^4 \), which is 81.

4. Is your answer negative?
   No, \( 3^{-4} \) is a positive number, \( \frac{1}{81} \).

5. What is -25 and how does it compare to (-2)^5?
   -25 is simply a negative number. \( (-2)^5 = -32 \). Comparing -25 and -32, -32 is less than -25 (i.e., -32 < -25).

6. Why are they different?
   The difference arises because -25 is a negative number that represents a fixed value, while \( (-2)^5 \) involves exponentiation of -2, resulting in a different negative outcome.

7. What is -(-3)^3?
   To visualize:

   • First, find \( (-3)^3 \): \[ (-3) \times (-3) \times (-3) = -27 \]
   • Therefore, \(-(-3)^3 = -(-27) = 27\).

8. What do you think of when you hear the word “exponents”?
   When I think of exponents, I think of multiplication that is repeated a number of times based on the exponent. It applies primarily in the context of powers.

9. What operation? How is it different?
   The operation is multiplication but applied multiple times. It is different from simple multiplication because the exponent indicates how many times the base is used in the multiplication.

10. What is the difference between \( 3(2) \) and \( 3^2 \)?

    • \( 3(2) \) means \( 3 \times 2 = 6 \).
    • \( 3^2 \) means \( 3 \times 3 = 9 \). Thus, \( 3(2) \) and \( 3^2 \) yield different results based on the operations performed.

What Do I Need Help With?

This is a space for you to request clarification on concepts, problems, or any specific topics you find challenging regarding exponents or related mathematical principles. Please feel free to ask, and I'll be glad to assist!