Sure! I'll provide you with a set of problems that cover the topics of rational numbers, linear equations in one variable, squares and square roots, and cubes and cube roots. These problems are suitable for a Class 8 exam format, and they can be solved using the formulas and concepts from each chapter. I'll include a mix of easy, moderate, and challenging problems to give a well-rounded set.

Exam-style Questions for Class 8:

Section A: Rational Numbers (2 Marks Each)

1. Find the multiplicative inverse of the rational number $-\frac{7}{15}$.

2. Simplify the expression:
   $\frac{3}{5} + \left( \frac{-4}{15} \times \frac{5}{6} \right)$

3. Divide:
   $\frac{7}{12} \div \left( -\frac{14}{9} \right)$

4. Find the value of:
   $\frac{-5}{8} + \frac{9}{16} - \frac{3}{4}$

Section B: Linear Equations in One Variable (3 Marks Each)

5. Solve for $x$:
   $5x - 3 = 2x + 9$

6. If $\frac{2x}{3} - 4 = 10$, find the value of $x$.

7. A number is doubled and then 7 is subtracted from it. The result is 13. Find the number.

8. Solve for $y$:
   $\frac{y}{2} + 3 = \frac{5y}{6} - 1$

Section C: Square and Square Roots (4 Marks Each)

9. Find the square root of $2025$ using prime factorization.

10. If $x^2 = 144$, find the value of $x$.

11. Evaluate:
    $\sqrt{16} + \sqrt{81} - \sqrt{49}$

12. Find the smallest number by which $180$ should be multiplied to make it a perfect square. Also, find the square root of the resulting number.

Section D: Cube and Cube Roots (5 Marks Each)

13. Find the cube root of $512$.

14. Evaluate:
    [ \sqrt[3]{-125} + \sqrt[3]{64} ]

15. If the volume of a cube is $729 \, \text{cm}^3$, find the length of its side.

16. Find the smallest number by which $48$ must be multiplied so that the product is a perfect cube. Also, find the cube root of the resulting number.

Answers Key (Brief):

1. Multiplicative inverse of $-\frac{7}{15}$ is $-\frac{15}{7}$.
2. Simplify: $\frac{3}{5} - \frac{2}{9} = \frac{17}{45}$.
3. Division: $\frac{7}{12} \times \frac{-9}{14} = -\frac{3}{8}$.
4. Simplify: $\frac{-5}{8} + \frac{9}{16} - \frac{3}{4} = -\frac{5}{16}$.
5. Solve: $x = 4$.
6. Solve: $x = 21$.
7. Number is $10$.
8. Solve: $y = 12$.
9. Square root of $2025$ is $45$.
10. $x = 12$ or $x = -12$.
11. Evaluate: $4 + 9 - 7 = 6$.
12. Multiply $180$ by $5$; square root is $30$.
13. Cube root of $512$ is $8$.
14. Evaluate: $-5 + 4 = -1$.
15. Side length is $9 \, \text{cm}$.
16. Multiply $48$ by $6$; cube root is $24$.

Would you like a more detailed explanation for any of these questions or concepts?

Follow-Up Questions:

1. What are some methods to solve linear equations in one variable?
2. How do you find the cube root of a large number without using a calculator?
3. What are some common mistakes when solving problems involving square roots?
4. Can rational numbers be both positive and negative? Provide examples.
5. How do you determine if a number is a perfect cube?

Tip:

Always double-check your final answer by substituting it back into the original equation or problem to ensure it satisfies all conditions.