Let's solve the algebraic expressions in the image step by step.

Question 3:

Part (a):

The given expression is:

$\frac{p + 5}{p - 5} \div \frac{p + 5}{p^2 - 25}$

To divide fractions, multiply by the reciprocal of the second fraction. The division becomes multiplication:

$\frac{p + 5}{p - 5} \times \frac{p^2 - 25}{p + 5}$

Now, recognize that $p^2 - 25$ is a difference of squares:

$p^2 - 25 = (p + 5)(p - 5)$

So, the expression becomes:

$\frac{p + 5}{p - 5} \times \frac{(p + 5)(p - 5)}{p + 5}$

Now cancel out the common factors:

• $p + 5$ in the numerator and denominator cancels out.
• $p - 5$ also cancels out.

After cancellation, the simplified expression is:

$1$

Part (b):

The given expression is:

$\frac{4c^4 d^{-3}}{2d^{-3}} \times \left( \frac{4c^5 d^3}{2c^4} \right)^{-2}$

First, simplify the first part:

$\frac{4c^4 d^{-3}}{2d^{-3}} = 2c^4$

Now handle the second part with the exponent $-2$. First simplify the fraction inside the parentheses:

$\frac{4c^5 d^3}{2c^4} = 2c d^3$

Now raise this to the power of $-2$:

$(2c d^3)^{-2} = \frac{1}{4c^2 d^6}$

Now multiply the simplified first part by the second part:

$2c^4 \times \frac{1}{4c^2 d^6} = \frac{2c^4}{4c^2 d^6} = \frac{c^2}{2d^6}$

Thus, the simplified expression is:

$\frac{c^2}{2d^6}$

Final Answers:

• Part (a): $1$
• Part (b): $\frac{c^2}{2d^6}$

Would you like more details on any of the steps or have any other questions?

Here are five related questions to deepen your understanding:

1. How would the solution change if the division symbol was replaced by a multiplication symbol in part (a)?
2. Can you explain why canceling terms in a fraction is valid?
3. How would you simplify expressions involving higher negative powers?
4. What role do the properties of exponents play in simplifying part (b)?
5. How would factoring help in simplifying more complex rational expressions?

Tip: Always look for opportunities to factor or cancel terms early on to simplify algebraic expressions faster.