Question

Myrna was asked to create an equivalent form of the rational expression 56k^7-14k^3+14/7k^3 Which of the following is a correct equivalent form?

Answer 1

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.

1. **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:

\[
\frac{14(4k^7 - k^3 + 1)}{7k^3}
\]

2. **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:

\[
= 2 \cdot \frac{(4k^7 - k^3 + 1)}{k^3}
\]

3. **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:

\[
2 \left( 4k^4 - 1 + \frac{1}{k^3} \right)
\]

4. **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

\[
8k^4 - 2 + \frac{2}{k^3}
\]