To simplify the rational expression \( \frac{56k^7 - 14k^3 + 14}{7k^3} \), we can start by factoring out the numerator and then dividing each term by the denominator.
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Factor the numerator:
The numerator is \( 56k^7 - 14k^3 + 14 \). We can factor out the greatest common factor (GCF) from the first two terms:
\[ 56k^7 - 14k^3 + 14 = 14(4k^7 - k^3 + 1) \]
This gives us:
\[ \frac{14(4k^7 - k^3 + 1)}{7k^3} \]
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Divide by the denominator:
Now, we can simplify the expression further by dividing both the numerator and denominator by 7:
\[ = \frac{14}{7} \cdot \frac{(4k^7 - k^3 + 1)}{k^3} \]
Simplifying \( \frac{14}{7} \) gives us 2:
\[ = 2 \cdot \frac{(4k^7 - k^3 + 1)}{k^3} \]
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Break down the fraction:
Now we can separate \( \frac{(4k^7 - k^3 + 1)}{k^3} \):
\[ = 2 \left( \frac{4k^7}{k^3} - \frac{k^3}{k^3} + \frac{1}{k^3} \right) \]
Simplifying each term:
- \( \frac{4k^7}{k^3} = 4k^{7-3} = 4k^4 \)
- \( \frac{k^3}{k^3} = 1 \)
- \( \frac{1}{k^3} = k^{-3} \)
So the expression becomes:
\[ 2 \left( 4k^4 - 1 + \frac{1}{k^3} \right) \]
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Final equivalent form:
We can then express it as:
\[ 8k^4 - 2 + \frac{2}{k^3} \]
So, an equivalent form of the rational expression \( \frac{56k^7 - 14k^3 + 14}{7k^3} \) is
\[ 8k^4 - 2 + \frac{2}{k^3} \]