Question

Myrna was asked to create an equivalent form of the rational expression 56k7−14k3+147k3. Which of the following is a correct equivalent form?(1 point)
Responses

8k4−2+2k3
8 superscript k 4 baseline minus 2 plus Start Fraction 2 over k cubed End Fraction

8k4+2k3
8 superscript k 4 baseline plus Start Fraction 2 over k cubed End Fraction

8k4−2k+2k3
8 superscript k 4 baseline minus 2 k plus Start Fraction 2 over k cubed End Fraction

8k10−2k6+2k3

Answer 1

To create an equivalent form of the rational expression \( 56k^7 - 14k^3 + 147k^3 \), we can simplify and combine like terms first.

1. Combine the like terms \( -14k^3 + 147k^3 \):
\[
-14k^3 + 147k^3 = 133k^3
\]

So, the expression simplifies to:
\[
56k^7 + 133k^3
\]

2. We can factor out the greatest common factor. The GCF of the coefficients 56 and 133 is 1, but we can factor out \( k^3 \):
\[
= k^3(56k^4 + 133)
\]

Now, we evaluate the provided options to see if any match an equivalent form:

1. \( 8k^4 - 2 + \frac{2}{k^3} \)
2. \( 8k^4 + \frac{2}{k^3} \)
3. \( 8k^4 - 2k + \frac{2}{k^3} \)
4. \( 8k^{10} - 2k^6 + 2k^3 \)

None of the options seem to correctly directly correlate with our factored expression \( k^3(56k^4 + 133) \).

Let’s rewrite \( 56 \) in terms of a coefficient of \( 8 \):
\[
56 = 8 \times 7
\]
Substituting this back, we get:
\[
k^3(8 \times 7 k^4 + 133) = 8k^4 + \frac{k^3 \times 133}{k^3} = 8k^4 + 133
\]

Given all this analysis, the correct equivalent form can be inferred is closest to the structure of those provided.

**Correct Answer**: The expression \( 56k^7 - 14k^3 + 147k^3 \) does not exactly match any standardized form of the options. The original expression simplifies primarily to \( k^3(56k^4 + 133) \), reaffirming those coefficients.

However, if the options provided must be selected, among them:
- **\( 8k^4 + 2k^3 \)** is the most rational approximation since \( 133k^3 \approx 2k^3 \) (only if assuming there was a miscalculation in listing out of context).

In conclusion:
- The best candidate, considering simplification and trailing composite coefficients that best expresses equivalency due to misunderstanding, might be **Option 2: \( 8k^4 + 2k^3 \)** if we adapt from \( 133 \) to conclusions given.

To clarify further, detailed evaluation or expression should nonetheless frame exact correctness within the setups laid out.