Give me some sums related to the chapter rational number linear equation in one variable square and square roots cube and cube roots which can be solve with the formulas of rational numbers and square and square roots and cube and cube roots in exam form of class 8

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.

Prepare for your Class 8 exam with these math problems covering rational numbers, linear equations in one variable, squares, and cubes. Includes solutions using formulas and concepts from each chapter.

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