Question
Yes the properties of exponents to determine whether the equation 10 ^-1/3=-^3 square root 10
Options
False beacayse the negative exponent rule requires that the number be rewritten as a quotient not as a negative value
False because according to the negative exponent rule the equation should be 10^-1/3 = -1/^3 square root 10
True because the negative sign moves to the front when writing an equivalent expression
True because the negative exponent rule requires that the number be rewritten as a negative value
Options
False beacayse the negative exponent rule requires that the number be rewritten as a quotient not as a negative value
False because according to the negative exponent rule the equation should be 10^-1/3 = -1/^3 square root 10
True because the negative sign moves to the front when writing an equivalent expression
True because the negative exponent rule requires that the number be rewritten as a negative value
Answers
Answered by
GPT-4o mini
To determine whether the equation \( 10^{-1/3} = -\sqrt[3]{10} \) is true or false, let's analyze both sides.
1. **Left Side:**
\( 10^{-1/3} \) can be rewritten using the negative exponent rule:
\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]
2. **Right Side:**
The right side is \( -\sqrt[3]{10} \).
Now we can directly compare the two sides:
- The left side is \( \frac{1}{\sqrt[3]{10}} \).
- The right side is \( -\sqrt[3]{10} \).
These two expressions are not equal.
Therefore, the equation \( 10^{-1/3} = -\sqrt[3]{10} \) is **False**.
Now, let's consider the provided options:
1. False because the negative exponent rule requires that the number be rewritten as a quotient not as a negative value. (This is somewhat correct but does not capture the reasoning fully).
2. False because according to the negative exponent rule the equation should be \( 10^{-1/3} = -\frac{1}{3} \sqrt{10} \) (This is incorrect as it doesn't accurately represent the left side).
3. True because the negative sign moves to the front when writing an equivalent expression (This is incorrect).
4. True because the negative exponent rule requires that the number be rewritten as a negative value (This is incorrect).
The best answer from the options is the first one, emphasizing that the negative exponent rule requires rewriting as a quotient. However, please note that the reasoning could have been more precise about the equivalence of both sides of the equation. Therefore, the overall conclusion is that the equation is **False**.
1. **Left Side:**
\( 10^{-1/3} \) can be rewritten using the negative exponent rule:
\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]
2. **Right Side:**
The right side is \( -\sqrt[3]{10} \).
Now we can directly compare the two sides:
- The left side is \( \frac{1}{\sqrt[3]{10}} \).
- The right side is \( -\sqrt[3]{10} \).
These two expressions are not equal.
Therefore, the equation \( 10^{-1/3} = -\sqrt[3]{10} \) is **False**.
Now, let's consider the provided options:
1. False because the negative exponent rule requires that the number be rewritten as a quotient not as a negative value. (This is somewhat correct but does not capture the reasoning fully).
2. False because according to the negative exponent rule the equation should be \( 10^{-1/3} = -\frac{1}{3} \sqrt{10} \) (This is incorrect as it doesn't accurately represent the left side).
3. True because the negative sign moves to the front when writing an equivalent expression (This is incorrect).
4. True because the negative exponent rule requires that the number be rewritten as a negative value (This is incorrect).
The best answer from the options is the first one, emphasizing that the negative exponent rule requires rewriting as a quotient. However, please note that the reasoning could have been more precise about the equivalence of both sides of the equation. Therefore, the overall conclusion is that the equation is **False**.