Select the correct answer to the following equation: x5/4=243

Question

.
3
3 - why is this incorrect

27
27 - no response given

no solution
no solution - no response given

81 why is this correct

Answer 1

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.

Answer 2

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

Answer 3

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 \( x^{5/4} = 243 \).