Question
Which statements are true regarding the expression below? Check all that apply.
8 Superscript negative 1 Baseline times 8 Superscript negative 3 Baseline times 8
The last factor, 8, with no exponent shown is equivalent to 8 Superscript 0.
The sum of the exponents is Negative 3.
The value of the expression is Negative 512.
The value of the expression is StartFraction 1 Over 512 EndFraction.
An equivalent expression is 8 Superscript 7 Baseline times 8 Superscript negative 10.
An equivalent expression is 8 Superscript negative 2 Baseline times 8 Superscript negative 2.
8 Superscript negative 1 Baseline times 8 Superscript negative 3 Baseline times 8
The last factor, 8, with no exponent shown is equivalent to 8 Superscript 0.
The sum of the exponents is Negative 3.
The value of the expression is Negative 512.
The value of the expression is StartFraction 1 Over 512 EndFraction.
An equivalent expression is 8 Superscript 7 Baseline times 8 Superscript negative 10.
An equivalent expression is 8 Superscript negative 2 Baseline times 8 Superscript negative 2.
Answers
GPT-4o mini
To analyze the expression \(8^{-1} \times 8^{-3} \times 8\), we can follow some mathematical principles. Here are the steps to evaluate the expression and the truth of each statement:
1. **Rewrite the last factor**: The last factor \(8\) can be written as \(8^1\), not \(8^0\). The exponent \(0\) would represent the value of \(1\), while \(8^1\) equals \(8\).
- **Statement**: "The last factor, 8, with no exponent shown is equivalent to \(8^0\)."
- **Truth**: **False**
2. **Sum of the exponents**: The exponents in the expression are \(-1\), \(-3\), and \(1\).
\[
-1 + (-3) + 1 = -1 - 3 + 1 = -3
\]
- **Statement**: "The sum of the exponents is \(-3\)."
- **Truth**: **True**
3. **Value of the expression**: We can simplify the expression using the property of exponents:
\[
8^{-1} \times 8^{-3} \times 8^1 = 8^{-1 - 3 + 1} = 8^{-3} = \frac{1}{8^3} = \frac{1}{512}
\]
- **Statement**: "The value of the expression is \(-512\)."
- **Truth**: **False**
- **Statement**: "The value of the expression is \(\frac{1}{512}\)."
- **Truth**: **True**
4. **Equivalent expressions**:
- To check the expression \(8^7 \times 8^{-10}\):
\[
8^7 \times 8^{-10} = 8^{7 - 10} = 8^{-3}
\]
- **Statement**: "An equivalent expression is \(8^7 \times 8^{-10}\)."
- **Truth**: **True**
- For \(8^{-2} \times 8^{-2}\):
\[
8^{-2} \times 8^{-2} = 8^{-2 - 2} = 8^{-4}
\]
- **Statement**: "An equivalent expression is \(8^{-2} \times 8^{-2}\)."
- **Truth**: **False**
In conclusion, the statements that are true are:
- The sum of the exponents is \(-3\).
- The value of the expression is \(\frac{1}{512}\).
- An equivalent expression is \(8^7 \times 8^{-10}\).
1. **Rewrite the last factor**: The last factor \(8\) can be written as \(8^1\), not \(8^0\). The exponent \(0\) would represent the value of \(1\), while \(8^1\) equals \(8\).
- **Statement**: "The last factor, 8, with no exponent shown is equivalent to \(8^0\)."
- **Truth**: **False**
2. **Sum of the exponents**: The exponents in the expression are \(-1\), \(-3\), and \(1\).
\[
-1 + (-3) + 1 = -1 - 3 + 1 = -3
\]
- **Statement**: "The sum of the exponents is \(-3\)."
- **Truth**: **True**
3. **Value of the expression**: We can simplify the expression using the property of exponents:
\[
8^{-1} \times 8^{-3} \times 8^1 = 8^{-1 - 3 + 1} = 8^{-3} = \frac{1}{8^3} = \frac{1}{512}
\]
- **Statement**: "The value of the expression is \(-512\)."
- **Truth**: **False**
- **Statement**: "The value of the expression is \(\frac{1}{512}\)."
- **Truth**: **True**
4. **Equivalent expressions**:
- To check the expression \(8^7 \times 8^{-10}\):
\[
8^7 \times 8^{-10} = 8^{7 - 10} = 8^{-3}
\]
- **Statement**: "An equivalent expression is \(8^7 \times 8^{-10}\)."
- **Truth**: **True**
- For \(8^{-2} \times 8^{-2}\):
\[
8^{-2} \times 8^{-2} = 8^{-2 - 2} = 8^{-4}
\]
- **Statement**: "An equivalent expression is \(8^{-2} \times 8^{-2}\)."
- **Truth**: **False**
In conclusion, the statements that are true are:
- The sum of the exponents is \(-3\).
- The value of the expression is \(\frac{1}{512}\).
- An equivalent expression is \(8^7 \times 8^{-10}\).