Let's break down and solve each part of the question.

Part (a): Simplify the expression

$\frac{x - 1}{5(x - 1)^2}$

Step-by-step solution:

1. Cancel common terms: The term $(x - 1)$ appears in both the numerator and denominator.

   • The numerator has $x - 1$.
   • The denominator has $(x - 1)^2$, which is $(x - 1) \times (x - 1)$.

   You can cancel one $x - 1$ term from both the numerator and the denominator: $\frac{(x - 1)}{5(x - 1)^2} = \frac{1}{5(x - 1)}$

Thus, the simplified expression is: $\frac{1}{5(x - 1)}$

Part (b): Factorize fully

$50 - 2y^2$

Step-by-step solution:

1. Factor out the greatest common factor (GCF): The GCF of $50$ and $2y^2$ is $2$. Factor out $2$: $50 - 2y^2 = 2(25 - y^2)$

2. Recognize a difference of squares: The expression inside the parentheses, $25 - y^2$, is a difference of squares: $25 - y^2 = (5 - y)(5 + y)$

Thus, the fully factorized form is: $2(5 - y)(5 + y)$

Do you want further details on any of the steps or have any questions?

5 Related Questions:

1. What are the general rules for canceling terms in algebraic expressions?
2. How can you identify a difference of squares quickly?
3. Why is it important to factor out the greatest common factor first in factoring problems?
4. Can the process of simplifying and factoring be applied to higher-order polynomial expressions?
5. How do you approach expressions with more complex denominators when simplifying?

Tip:

Always check for common factors before moving on to more complex factoring techniques; this can make the process much easier.