Question
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 ?
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 ?
Answers
Answered by
GPT-4o mini
### Vocabulary Terms
1. **Exponent**: An exponent is a mathematical notation that indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base. For example, in \( 2^3 \), 2 is the base, and 3 is the exponent.
**Illustration**: \( 3^4 = 3 \times 3 \times 3 \times 3 \)
2. **Base**: The base is the number that is being multiplied when using an exponent. In the expression \( 5^2 \), 5 is the base.
3. **Standard form vs. Exponent form**:
- **Standard form**: This is the conventional way of writing numbers. For example, \( 4^3 = 64 \).
- **Exponent form**: This represents numbers using exponents, e.g., instead of writing \( 4 \times 4 \times 4 \), we write \( 4^3 \).
---
### Problems
1. **What does \( x^4 \) mean?**
- \( x^4 \) means \( x \times x \times x \times x \). It signifies that the variable \( x \) is multiplied by itself 4 times.
2. **What operation are you doing?**
- The operation is multiplication.
3. **What does \( 3^{-4} \) mean?**
- \( 3^{-4} \) is the reciprocal of \( 3^4 \). It can be written as \( \frac{1}{3^4} \), which equals \( \frac{1}{81} \).
4. **Is your answer negative?**
- No, \( 3^{-4} \) is not a negative number; it results in a positive fraction, \( \frac{1}{81} \).
5. **What is \( -25 \) and how does it compare to \( (-2)^5 \)?**
- \( -25 \) is simply negative twenty-five.
- \( (-2)^5 \) calculates as \( -2 \times -2 \times -2 \times -2 \times -2 \), which equals \( -32 \).
- So, \( -25 \) and \( (-2)^5 \) are different; \( -25 \) is not equal to \( -32 \).
6. **Why are they different?**
- They are different because they come from different bases. \( (-2)^5 \) is evaluating the exponentiation of \( -2 \), resulting in \( -32 \), while \( -25 \) is a constant negative value.
7. **What is \( -(-3)^3 \)?**
- Write it out in long hand:
\[
-(-3) \times -3 \times -3 = -(-27) = 27
\]
8. **What do you think of when you hear the word “exponents”? What operation? How is it different?**
- When I hear “exponents,” I think of powers and multiplication. Exponents are a shorthand way of denoting repeated multiplication, which is different from simple multiplication (like addition or subtraction) because it deals with how many times a number is multiplied by itself rather than just combining numbers.
9. **What is the difference between \( 3(2) \) and \( 3^2 \)?**
- \( 3(2) \) means \( 3 \times 2 = 6 \).
- \( 3^2 \) means \( 3 \times 3 = 9 \).
- The difference lies in the operations; \( 3(2) \) is a multiplication of \( 3 \) and \( 2 \), while \( 3^2 \) is \( 3 \) multiplied by itself.
---
This should cover all the vocabulary and problems thoroughly, along with clear explanations and illustrations. Let me know if you need further assistance!
1. **Exponent**: An exponent is a mathematical notation that indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base. For example, in \( 2^3 \), 2 is the base, and 3 is the exponent.
**Illustration**: \( 3^4 = 3 \times 3 \times 3 \times 3 \)
2. **Base**: The base is the number that is being multiplied when using an exponent. In the expression \( 5^2 \), 5 is the base.
3. **Standard form vs. Exponent form**:
- **Standard form**: This is the conventional way of writing numbers. For example, \( 4^3 = 64 \).
- **Exponent form**: This represents numbers using exponents, e.g., instead of writing \( 4 \times 4 \times 4 \), we write \( 4^3 \).
---
### Problems
1. **What does \( x^4 \) mean?**
- \( x^4 \) means \( x \times x \times x \times x \). It signifies that the variable \( x \) is multiplied by itself 4 times.
2. **What operation are you doing?**
- The operation is multiplication.
3. **What does \( 3^{-4} \) mean?**
- \( 3^{-4} \) is the reciprocal of \( 3^4 \). It can be written as \( \frac{1}{3^4} \), which equals \( \frac{1}{81} \).
4. **Is your answer negative?**
- No, \( 3^{-4} \) is not a negative number; it results in a positive fraction, \( \frac{1}{81} \).
5. **What is \( -25 \) and how does it compare to \( (-2)^5 \)?**
- \( -25 \) is simply negative twenty-five.
- \( (-2)^5 \) calculates as \( -2 \times -2 \times -2 \times -2 \times -2 \), which equals \( -32 \).
- So, \( -25 \) and \( (-2)^5 \) are different; \( -25 \) is not equal to \( -32 \).
6. **Why are they different?**
- They are different because they come from different bases. \( (-2)^5 \) is evaluating the exponentiation of \( -2 \), resulting in \( -32 \), while \( -25 \) is a constant negative value.
7. **What is \( -(-3)^3 \)?**
- Write it out in long hand:
\[
-(-3) \times -3 \times -3 = -(-27) = 27
\]
8. **What do you think of when you hear the word “exponents”? What operation? How is it different?**
- When I hear “exponents,” I think of powers and multiplication. Exponents are a shorthand way of denoting repeated multiplication, which is different from simple multiplication (like addition or subtraction) because it deals with how many times a number is multiplied by itself rather than just combining numbers.
9. **What is the difference between \( 3(2) \) and \( 3^2 \)?**
- \( 3(2) \) means \( 3 \times 2 = 6 \).
- \( 3^2 \) means \( 3 \times 3 = 9 \).
- The difference lies in the operations; \( 3(2) \) is a multiplication of \( 3 \) and \( 2 \), while \( 3^2 \) is \( 3 \) multiplied by itself.
---
This should cover all the vocabulary and problems thoroughly, along with clear explanations and illustrations. Let me know if you need further assistance!
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