Sure! Let's break this down step by step, starting with the arrangement of the values from least to greatest.
1. Arranging Values
Given:
- A = 3
- B = -2
- C = 5
- D = 0
- E = -1
- F = 4
- G = -3
Step 1: Identify the Values
Now let's list the values:
- A (3)
- B (-2)
- C (5)
- D (0)
- E (-1)
- F (4)
- G (-3)
Step 2: Order Them
To arrange these from least to greatest, we need to compare the numbers:
- The least value is G (-3)
- Next is B (-2)
- Then E (-1)
- D is next (0)
- Followed by A (3)
- Then F (4)
- Finally, C (5)
Ordered List:
G, B, E, D, A, F, C
(-3, -2, -1, 0, 3, 4, 5)
2. Vocabulary Terms
Exponent: An exponent is a mathematical notation that indicates how many times a number (the base) is multiplied by itself.
Base: The base is the number that is being raised to a power.
Standard form vs. Exponent form:
- Standard form is the usual way to write a number, for example, 1000.
- Exponent form expresses a number using exponents, for example, \(10^3\) represents \(10 \times 10 \times 10\), or 1000.
3. Problems
1. What does \(x^4\) mean?
\(x^4\) means that \(x\) is multiplied by itself four times: \(x \times x \times x \times x\).
2. What operation are you doing?
You are performing multiplication of the base \(x\) for four times.
3. What does \(3^{-4}\) mean?
\(3^{-4}\) means \(\frac{1}{3^4}\), which is \(\frac{1}{81}\).
4. Is your answer negative?
No, because raising a positive number (3) to any exponent, including a negative exponent, results in a positive value.
5. What is \(-25\) and how does it compare to \((-2)^5\)?
- \(-25\) is a negative number.
- \((-2)^5 = -32\) because multiplying \(-2\) by itself five times gives \(-2 \times -2 \times -2 \times -2 \times -2 = -32\).
They are different because \(-25\) is a negative number, while \((-2)^5\) results in a larger negative number (-32).
6. What is \(-(-3)^3\)?
To write it out in longhand:
\(-(-3)^3 = -(-3 \times -3 \times -3) = -(-27) = 27\).
7. What do you think of when you hear the word “exponents”? What operation?
When I hear "exponents," I think of multiplication, specifically multiplying a base number by itself a certain number of times.
8. How is it different?
Exponents involve repeated multiplication rather than just multiplication. For example, \(a \times b\) is a single multiplication, while \(a^b\) (where b is an integer) represents multiple multiplications of \(a\).
Example: What is the difference between \(3(2)\) and \(3^2\)?
- \(3(2) = 6\) (multiply 3 by 2)
- \(3^2 = 9\) (3 multiplied by itself: \(3 \times 3\))
4. What do I need help with?
In this section, you can write down any concepts or problems that you're struggling with. For example, are you unclear about understanding exponents, operations, or comparisons? Don't hesitate to outline specific topics so that you can focus your study efforts more effectively!
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