To create an equivalent form of the rational expression \( 56k^7 - 14k^3 + 147k^3 \), we can simplify and combine like terms first.
-
Combine the like terms \( -14k^3 + 147k^3 \): \[ -14k^3 + 147k^3 = 133k^3 \]
So, the expression simplifies to: \[ 56k^7 + 133k^3 \]
-
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:
- \( 8k^4 - 2 + \frac{2}{k^3} \)
- \( 8k^4 + \frac{2}{k^3} \)
- \( 8k^4 - 2k + \frac{2}{k^3} \)
- \( 8k^{10} - 2k^6 + 2k^3 \)
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.