Questions: Simplify. [ fracfrac7u v^2frac5u^2+frac1v ]

Solution Steps

To simplify the given expression, we need to perform the following steps:

1. Simplify the numerator and the denominator separately.
2. For the numerator, it's already a single fraction: \(\frac{7}{uv^2}\).
3. For the denominator, find a common denominator for the terms \(\frac{5}{u^2}\) and \(\frac{1}{v}\).
4. Combine the terms in the denominator over the common denominator.
5. Divide the simplified numerator by the simplified denominator.

The numerator of the expression is given as: \[ \text{Numerator} = \frac{7}{uv^2} \]

The denominator consists of two fractions: \[ \text{Denominator} = \frac{5}{u^2} + \frac{1}{v} \] To combine these fractions, we find a common denominator, which is \(u^2v\): \[ \text{Denominator} = \frac{5v}{u^2v} + \frac{u^2}{u^2v} = \frac{5v + u^2}{u^2v} \]

Now we can express the entire fraction: \[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{7}{uv^2}}{\frac{5v + u^2}{u^2v}} = \frac{7}{uv^2} \cdot \frac{u^2v}{5v + u^2} \] This simplifies to: \[ \frac{7u}{v(5v + u^2)} \]

Final Answer