Question
Using the properties of exponents, which expression is equivalent to x^1/6⋅x^1/6
?(1 point)
Responses
x−−√
Start Root x End Root
x−−√^36
root index 36 Start Root x End Root
x−−√^3
root index 3 Start Root x End Root
1/3√x
?(1 point)
Responses
x−−√
Start Root x End Root
x−−√^36
root index 36 Start Root x End Root
x−−√^3
root index 3 Start Root x End Root
1/3√x
Answers
GPT-4o mini
To simplify the expression \( x^{1/6} \cdot x^{1/6} \), we can apply the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
Thus, we have:
\[
x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3}
\]
Now, we can express \( x^{1/3} \) in radical form.
Recall that \( x^{1/n} \) is equivalent to \( \sqrt[n]{x} \), which means:
\[
x^{1/3} = \sqrt[3]{x}
\]
Now let's analyze the provided responses:
1. **\( \sqrt{x} \)** (This is \( x^{1/2} \), not equivalent)
2. **\( \sqrt[36]{x} \)** (This is \( x^{1/36} \), not equivalent)
3. **\( \sqrt[3]{x} \)** (This is \( x^{1/3} \), equivalent)
4. **\( \frac{1}{3}\sqrt{x} \)** (This is also not equivalent)
The correct answer is:
**\( \sqrt[3]{x} \)**
Thus, we have:
\[
x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3}
\]
Now, we can express \( x^{1/3} \) in radical form.
Recall that \( x^{1/n} \) is equivalent to \( \sqrt[n]{x} \), which means:
\[
x^{1/3} = \sqrt[3]{x}
\]
Now let's analyze the provided responses:
1. **\( \sqrt{x} \)** (This is \( x^{1/2} \), not equivalent)
2. **\( \sqrt[36]{x} \)** (This is \( x^{1/36} \), not equivalent)
3. **\( \sqrt[3]{x} \)** (This is \( x^{1/3} \), equivalent)
4. **\( \frac{1}{3}\sqrt{x} \)** (This is also not equivalent)
The correct answer is:
**\( \sqrt[3]{x} \)**
Using the properties of exponents, which expression is equivalent to x^x/3/4
?(1 point)
Responses
x−−√^4
root index 4 Start Root x End Root
x^4
x superscript 4 baseline
x
x
1/4^√x
?(1 point)
Responses
x−−√^4
root index 4 Start Root x End Root
x^4
x superscript 4 baseline
x
x
1/4^√x
GPT-4o mini
The expression you provided, \( x^{x/3}/4 \), seems to have been formatted incorrectly. It appears there may be some confusion about whether you meant to write \( \frac{x^{x/3}}{4} \) or \( x^{\frac{x}{3} \cdot 4} \).
However, I will assume you meant \( x^{x/3} \) for simplicity and to find its equivalent form based on the options provided.
Assuming \( x^{x/3} \):
1. To simplify \( x^{x/3} \), we can write it using the property of exponents, but it cannot be simplified into a simpler radical or more basic exponent form without knowing specific values for \( x \).
Let's evaluate your provided options:
1. **\( \sqrt[4]{x} \)** (This is \( x^{1/4} \), not equivalent)
2. **\( x^4 \)** (This is not equivalent either)
3. **\( x \)** (This is not equivalent)
4. **\( \frac{1}{4} \sqrt{x} \)** (This is also not equivalent)
None of the provided options equally represent \( x^{x/3} \).
If there's a mistake in the transcription of the original expression, please clarify it, and I will be happy to help further!
However, I will assume you meant \( x^{x/3} \) for simplicity and to find its equivalent form based on the options provided.
Assuming \( x^{x/3} \):
1. To simplify \( x^{x/3} \), we can write it using the property of exponents, but it cannot be simplified into a simpler radical or more basic exponent form without knowing specific values for \( x \).
Let's evaluate your provided options:
1. **\( \sqrt[4]{x} \)** (This is \( x^{1/4} \), not equivalent)
2. **\( x^4 \)** (This is not equivalent either)
3. **\( x \)** (This is not equivalent)
4. **\( \frac{1}{4} \sqrt{x} \)** (This is also not equivalent)
None of the provided options equally represent \( x^{x/3} \).
If there's a mistake in the transcription of the original expression, please clarify it, and I will be happy to help further!
Using the properties of exponents, which expression is equivalent to xx34
?(1 point)
Responses
x−−√4
root index 4 Start Root x End Root
x4
x superscript 4 baseline
x
x
1x√4
?(1 point)
Responses
x−−√4
root index 4 Start Root x End Root
x4
x superscript 4 baseline
x
x
1x√4
GPT-4o mini
To simplify the expression \( x^{x/3} \), we will use properties of exponents.
Assuming \( xx34 \) is meant to be \( x^{x/3} \):
1. Let’s clarify that \( x^{x/3} \) represents \( x \) raised to the power of \( \frac{x}{3} \), and we cannot simplify it further without specific values for \( x \).
Now, let's evaluate your provided options:
1. **\( \sqrt[4]{x} \)** - This is equivalent to \( x^{1/4} \), not equivalent to \( x^{x/3} \).
2. **\( x^4 \)** - This is not equivalent to \( x^{x/3} \).
3. **\( x \)** - This is not equivalent to \( x^{x/3} \).
4. **\( \frac{1}{x}\sqrt{4} \)** - This is also not equivalent to \( x^{x/3} \).
Given that none of the provided responses match \( x^{x/3} \), I can conclude that none of the options are correct based on this interpretation.
If you meant a different expression, please provide clarification!
Assuming \( xx34 \) is meant to be \( x^{x/3} \):
1. Let’s clarify that \( x^{x/3} \) represents \( x \) raised to the power of \( \frac{x}{3} \), and we cannot simplify it further without specific values for \( x \).
Now, let's evaluate your provided options:
1. **\( \sqrt[4]{x} \)** - This is equivalent to \( x^{1/4} \), not equivalent to \( x^{x/3} \).
2. **\( x^4 \)** - This is not equivalent to \( x^{x/3} \).
3. **\( x \)** - This is not equivalent to \( x^{x/3} \).
4. **\( \frac{1}{x}\sqrt{4} \)** - This is also not equivalent to \( x^{x/3} \).
Given that none of the provided responses match \( x^{x/3} \), I can conclude that none of the options are correct based on this interpretation.
If you meant a different expression, please provide clarification!