Question

Solve the following equation with a rational exponent: 5x43=80.(1 point)
Responses

16
16

12
12

8
8

40.32

Answer 1

To solve the equation \( 5x^{43} = 80 \), we first isolate \( x^{43} \):

\[
x^{43} = \frac{80}{5}
\]

Calculating the right side gives:

\[
x^{43} = 16
\]

Next, we take the \( 43 \)rd root of both sides. This can also be expressed using a rational exponent:

\[
x = 16^{\frac{1}{43}}
\]

Now we need to find the numerical value of \( 16^{\frac{1}{43}} \). Since \( 16 = 2^4 \), we can rewrite it as:

\[
x = (2^4)^{\frac{1}{43}} = 2^{\frac{4}{43}}
\]

Using a calculator, we can approximate \( 16^{\frac{1}{43}} \):

\( 16^{\frac{1}{43}} \approx 1.139 \).

As this isn't one of the options you provided, it is likely that the values given in the responses correspond to more direct answers. The most straightforward exponentiation does not yield any rational options.

If we consider potential rounding or approximations, here are the provided response options:
- 16
- 12
- 8
- 40.32

Since none of these responses correspond to \( 16^{\frac{1}{43}} \), and since \( x^{43} = 16 \) leads us down this path, it's likely a misinterpretation of the original problem or its variable forms.

**Final Answer:**
Given the calculations performed, \( x = 16^{\frac{1}{43}} \), and the choices do not correspond to this finding; please double-check the problem statement if there are any additional constraints or if adjustments to the equation might yield different options.