Asked by adaleta
Select the correct answer to the following equation: x5/4=243
.
3
3 - why is this incorrect
27
27 - no response given
no solution
no solution - no response given
81 why is this correct
.
3
3 - why is this incorrect
27
27 - no response given
no solution
no solution - no response given
81 why is this correct
Answers
Answered by
adaleta
Select the correct answer to the following equation: x^5/4=243
.
Responses
3
3 - incorrect why is it wrong
27
27 - no response given
no solution
no solution - no response given
81
81 - not selected, this is the correct answer why is right
Score
.
Responses
3
3 - incorrect why is it wrong
27
27 - no response given
no solution
no solution - no response given
81
81 - not selected, this is the correct answer why is right
Score
Answered by
GPT-4o mini
Let's solve the equation \( x^{5/4} = 243 \).
To isolate \( x \), we need to eliminate the exponent \( \frac{5}{4} \) by raising both sides to the reciprocal of \( \frac{5}{4} \), which is \( \frac{4}{5} \):
\[
x = 243^{\frac{4}{5}}
\]
Now, first we need to determine what \( 243 \) is in terms of powers. We know that:
\[
243 = 3^5
\]
Thus, we can substitute \( 243 \) in the equation:
\[
x = (3^5)^{\frac{4}{5}}
\]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we have:
\[
x = 3^{5 \cdot \frac{4}{5}} = 3^4
\]
Now we can calculate \( 3^4 \):
\[
3^4 = 81
\]
Therefore, the solution to the equation \( x^{5/4} = 243 \) is:
\[
x = 81
\]
This confirms that the correct answer is \( 81 \), making the reasoning for this solution correct.
To isolate \( x \), we need to eliminate the exponent \( \frac{5}{4} \) by raising both sides to the reciprocal of \( \frac{5}{4} \), which is \( \frac{4}{5} \):
\[
x = 243^{\frac{4}{5}}
\]
Now, first we need to determine what \( 243 \) is in terms of powers. We know that:
\[
243 = 3^5
\]
Thus, we can substitute \( 243 \) in the equation:
\[
x = (3^5)^{\frac{4}{5}}
\]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we have:
\[
x = 3^{5 \cdot \frac{4}{5}} = 3^4
\]
Now we can calculate \( 3^4 \):
\[
3^4 = 81
\]
Therefore, the solution to the equation \( x^{5/4} = 243 \) is:
\[
x = 81
\]
This confirms that the correct answer is \( 81 \), making the reasoning for this solution correct.
Answered by
GPT-4o mini
Let's break down the equation \( x^{5/4} = 243 \) step by step and explain why the various responses are correct or incorrect.
1. **Starting Equation**:
\[
x^{5/4} = 243
\]
2. **Isolating \( x \)**:
To solve for \( x \), we can raise both sides to the reciprocal of the exponent \( \frac{5}{4} \), which is \( \frac{4}{5} \):
\[
x = 243^{\frac{4}{5}}
\]
3. **Expressing 243 as a Power**:
We know that \( 243 = 3^5 \). This allows us to rewrite our equation:
\[
x = (3^5)^{\frac{4}{5}}
\]
4. **Calculating the Exponent**:
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify this further:
\[
x = 3^{5 \cdot \frac{4}{5}} = 3^4
\]
5. **Calculating \( 3^4 \)**:
Next, we calculate \( 3^4 \):
\[
3^4 = 81
\]
6. **Final Result**:
Thus, \( x = 81 \) is the solution to the equation.
### Evaluating Responses:
- **Response: 3**:
This value is incorrect because \( 3 \) raised to any power won’t yield \( 243^{5/4} \). Specifically, \( 3^{5/4} \approx 2.5 \) which is much less than \( 243\).
- **Response: 27**:
This value is also incorrect. \( 27 \) can be expressed as \( 3^3 \), which does not satisfy the equation since \( 27^{5/4} \) is not equal to \( 243 \).
- **Response: no solution**:
This is incorrect because we have found a valid solution, which is \( 81 \). The equation does have a solution.
- **Response: 81**:
This is the correct answer. As shown in the calculation, \( x = 81 \) satisfies the original equation \( x^{5/4} = 243 \).
Therefore, the correct answer is \( 81 \) because it is the only value we found through calculations that correctly satisfies the equation \( x^{5/4} = 243 \).
1. **Starting Equation**:
\[
x^{5/4} = 243
\]
2. **Isolating \( x \)**:
To solve for \( x \), we can raise both sides to the reciprocal of the exponent \( \frac{5}{4} \), which is \( \frac{4}{5} \):
\[
x = 243^{\frac{4}{5}}
\]
3. **Expressing 243 as a Power**:
We know that \( 243 = 3^5 \). This allows us to rewrite our equation:
\[
x = (3^5)^{\frac{4}{5}}
\]
4. **Calculating the Exponent**:
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify this further:
\[
x = 3^{5 \cdot \frac{4}{5}} = 3^4
\]
5. **Calculating \( 3^4 \)**:
Next, we calculate \( 3^4 \):
\[
3^4 = 81
\]
6. **Final Result**:
Thus, \( x = 81 \) is the solution to the equation.
### Evaluating Responses:
- **Response: 3**:
This value is incorrect because \( 3 \) raised to any power won’t yield \( 243^{5/4} \). Specifically, \( 3^{5/4} \approx 2.5 \) which is much less than \( 243\).
- **Response: 27**:
This value is also incorrect. \( 27 \) can be expressed as \( 3^3 \), which does not satisfy the equation since \( 27^{5/4} \) is not equal to \( 243 \).
- **Response: no solution**:
This is incorrect because we have found a valid solution, which is \( 81 \). The equation does have a solution.
- **Response: 81**:
This is the correct answer. As shown in the calculation, \( x = 81 \) satisfies the original equation \( x^{5/4} = 243 \).
Therefore, the correct answer is \( 81 \) because it is the only value we found through calculations that correctly satisfies the equation \( x^{5/4} = 243 \).
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