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Let's work through the questions shown in the image.

**Question 6:**
- Given a rectangle with dimensions \( \frac{5}{2} \) cm and \( \sqrt{2} \) cm.

**Part (a):** Find the perimeter of the rectangle.
- The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2(l + w)
\]
where \( l \) is the length and \( w \) is the width.

Here, the length \( l = \frac{5}{2} \) cm and the width \( w = \sqrt{2} \) cm. So:
\[
P = 2\left(\frac{5}{2} + \sqrt{2}\right)
= 2\left(\frac{5}{2} + \frac{\sqrt{2}}{1}\right)
= 2 \times \left(\frac{5 + 2\sqrt{2}}{2}\right)
= 5 + 2\sqrt{2} \, \text{cm}
\]

Thus, the perimeter is \( 5 + 2\sqrt{2} \) cm.

**Part (b):** Find the area of the rectangle.
- The area \( A \) of a rectangle is given by:
\[
A = l \times w
\]
Substituting the given values:
\[
A = \frac{5}{2} \times \sqrt{2} = \frac{5\sqrt{2}}{2} \, \text{cm}^2
\]

Thus, the area is \( \frac{5\sqrt{2}}{2} \) cm².

---

**Question 7:** Simplify each expression in the form \( a\sqrt{3} \), where \( a \) is an integer.

**Part (a):** \( \sqrt{6} \times \sqrt{8} \)
\[
\sqrt{6} \times \sqrt{8} = \sqrt{6 \times 8} = \sqrt{48}
\]
Simplify \( \sqrt{48} \):
\[
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
\]

So the answer is \( 4\sqrt{3} \).

**Part (b):** \( \sqrt{27} + \sqrt{75} \)
\[
\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
\]
\[
\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
\]
Now, add them:
\[
3\sqrt{3} + 5\sqrt{3} = 8\sqrt{3}
\]

Thus, the answer is \( 8\sqrt{3} \).

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Do you want more details or have any questions?

Here are 5 related questions:
1. How do you convert a product of square roots into a simplified form?
2. Can the perimeter formula be used for any polygon? If not, how does it differ?
3. What are some real-world applications of finding the area of rectangles?
4. How do you handle square roots of non-perfect squares in simplifications?
5. How does the distributive property work when simplifying square roots?

**Tip:** Always break down square roots of products or sums into prime factors for easy simplification.

6. Shown below is a rectangle with sides 5/2 cm and √2 cm. (a) Find the perimeter of the rectangle. (b) Find the area of the rectangle. 7. Write each of these in the form a√3, where a is an integer. (a) √6 × √8 (b) √27 + √75

Learn how to calculate the perimeter and area of a rectangle with dimensions 5/2 cm and √2 cm. Also, simplify square root expressions like √6 × √8 and √27 + √75 into the form a√3, where a is an integer. Suitable for students in Grades 9-10.

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