Question

\(\mathrm{x}=\frac{\mathrm{y}}{\mathrm{y}^{\prime}}+\frac{1}{\mathrm{y}^{\prime 2}}\)

Short answer

Answer: The simplified form of the expression is \(x=\frac{y * y' + 1}{(y')^2}\).

Step-by-step solution

Find the LCD

The least common denominator is the lowest common multiple (LCM) of the denominators. In this case, the LCM of \(\mathrm{y}^{\prime}\) and \(\mathrm{y}^{\prime2}\) is \(\mathrm{y}^{\prime2}\).

Combine the fractions

We can now rewrite the fractions using the LCD and then combine them: \(\frac{\mathrm{y}*\mathrm{y}^{\prime}}{\mathrm{y}^{\prime}*\mathrm{y}^{\prime}} + \frac{1}{\mathrm{y}^{\prime2}} = \frac{\mathrm{y} * \mathrm{y}^{\prime} + 1}{\mathrm{y}^{\prime2}}\).

Final Simplification

The expression is now simplified, and there are no common factors to factorize further. So, the final simplified expression is: \(\mathrm{x}=\frac{\mathrm{y} * \mathrm{y}^{\prime} + 1}{\mathrm{y}^{\prime2}}\).

Key concepts

Least Common Denominator

Understanding the least common denominator (LCD) is essential in the field of integral calculus, especially when dealing with fractions. The LCD refers to the smallest number that all denominators can divide into without a remainder. It's the lowest common multiple (LCM) of the denominators. Identifying the LCD allows us to combine fractions with different denominators, and it's the first step towards simplification. In our exercise example, where the denominators are \(\mathrm{y}'\) and \(\mathrm{y}'^2\), the LCD is \(\mathrm{y}'^2\). That's because \(\mathrm{y}'^2\) is the smallest number that both \(\mathrm{y}'\) and \(\mathrm{y}'^2\) can divide into. This common ground enables the merging of fractions for further operations.

Combining Fractions

When fractions share a common denominator, they can be easily combined. This procedure involves rewriting each fraction with the LCD as the new denominator and then adding or subtracting the numerators as one would with whole numbers. In the given exercise, we rewrite the fractions \(\frac{\mathrm{y}}{\mathrm{y}'}\) and \(\frac{1}{\mathrm{y}'^2}\) so they have the LCD \(\mathrm{y}'^2\). This results in \(\frac{\mathrm{y} * \mathrm{y}'}{\mathrm{y}'^2}\) for the first fraction, since multiplying the numerator and the denominator of the first fraction by \(\mathrm{y}'\) does not change its value but gives it the common denominator needed to combine it with the second fraction. After rewriting, these fractions are added together by summing the numerators, leading to a single fractional expression \(\frac{\mathrm{y} * \mathrm{y}' + 1}{\mathrm{y}'^2}\). Combining fractions is a technique that not only simplifies expressions but also paves the way for easier integration or other calculus-related operations.

Simplifying Expressions

Simplifying expressions is a fundamental part of algebra and calculus. It entails reducing an expression to its most basic form, making it easier to understand and work with. Simplification can involve combining like terms, factoring, expanding expressions, or reducing fractions. After combining fractions, as seen in the previous steps, the result occasionally contains common factors in both the numerator and the denominator that can be simplified. In our example, the expression \(\frac{\mathrm{y} * \mathrm{y}' + 1}{\mathrm{y}'^2}\) does not have any common factors that allow for further simplification. In such cases, we've reached the expression's simplest form, indicating the cancellation of common factors is not possible. Recognizing when an expression cannot be simplified any further is just as crucial as the process of simplification itself. A simplified expression is the goal, as it typically makes the subsequent mathematical procedures more straightforward and less prone to error.