Let's solve the problem step by step.

Given:

$p = \sqrt{16 + 2\sqrt{48}}$ $q = \sqrt{16 - 2\sqrt{28}}$

We need to find:

a. $p + q$
b. $p \times q$

Step 1: Simplify $p$

$p = \sqrt{16 + 2\sqrt{48}}$

We can try to express the term inside the square root as a perfect square: $\sqrt{16 + 2\sqrt{48}} = \sqrt{a + 2\sqrt{b}}$

Let's express $\sqrt{16 + 2\sqrt{48}}$ in the form $\sqrt{(x + \sqrt{y})^2}$.

Expanding $(x + \sqrt{y})^2$ gives: $(x + \sqrt{y})^2 = x^2 + 2x\sqrt{y} + y$ Now, compare this with the original expression $16 + 2\sqrt{48}$.

We need: $x^2 + y = 16$ $2x\sqrt{y} = 2\sqrt{48}$

From $2x\sqrt{y} = 2\sqrt{48}$, we can cancel the 2: $x\sqrt{y} = \sqrt{48}$

Since $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$, we have: $x\sqrt{y} = 4\sqrt{3}$

Let $x = 4$ and $y = 3$, which satisfies the equation $x\sqrt{y} = 4\sqrt{3}$.

Now check if $x^2 + y = 16$: $4^2 + 3 = 16 + 3 = 19$

It seems there was an error in the selection. Let's try another pair of values: If we set $x = 2\sqrt{3}$ and $y = 4$:

Then: $2\sqrt{3} \times \sqrt{4} = 4\sqrt{3}$

Now $y = 4$, so: $x^2 = 12$

But this leads to: $x^2 + y = 12 + 4 = 16$

So: $p = 4 + \sqrt{3}$

Step 2: Simplify $q$

$q = \sqrt{16 - 2\sqrt{28}}$

Following a similar approach: We assume: $q = x - \sqrt{y}$

Now: $16 - 2\sqrt{28} = \sqrt{(x - \sqrt{y})^2}$

From earlier, trying with $x$ and $y$: $q = 4 - \sqrt{3}$

Step 3: Solve for $p + q$

$p + q = (4 + \sqrt{3}) + (4 - \sqrt{3})$ $p + q = 8$

Step 4: Solve for $p \times q$

$p \times q = (4 + \sqrt{3})(4 - \sqrt{3})$

Using the difference of squares: $p \times q = 4^2 - (\sqrt{3})^2$ $p \times q = 16 - 3 = 13$

Final Answers:

a. $p + q = 8$
b. $p \times q = 13$

Would you like any more details or have any questions? Here are some related questions for further exploration:

1. How do you simplify expressions involving square roots?
2. What are some other methods for solving radicals without expansion?
3. What is the significance of the difference of squares in simplifying expressions?
4. Can all expressions inside square roots be represented as perfect squares?
5. How would you solve similar problems with higher-order roots?
6. What happens when you encounter a non-rational radical expression?
7. How can you check your work when simplifying square roots?
8. Are there alternative approaches to simplify the given expressions?

Tip: Always double-check your selected values for simplification by plugging them back into the original equation to ensure they satisfy all conditions.