Express of the rational numbers in standard form: `(i)(-12)/(30)" " (ii) (-14)/(49) " " (iii) (-24)/(64) " " (iv) (-36)/(-63)`

Answer

To express the given rational numbers in standard form, we need to simplify each fraction by finding the greatest common divisor (GCD) of the numerator and denominator and then dividing both by that GCD.

### Step-by-Step Solution:

**(i) Simplifying \(-\frac{12}{30}\)**

1. **Find the GCD of 12 and 30.**
- The factors of 12: 1, 2, 3, 4, 6, 12
- The factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- The common factors are 1, 2, 3, 6. The greatest common factor is 6.

2. **Divide both the numerator and denominator by the GCD (6).**
- \(-\frac{12 \div 6}{30 \div 6} = -\frac{2}{5}\)

**Standard form: \(-\frac{2}{5}\)**

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**(ii) Simplifying \(-\frac{14}{49}\)**

1. **Find the GCD of 14 and 49.**
- The factors of 14: 1, 2, 7, 14
- The factors of 49: 1, 7, 49
- The common factors are 1, 7. The greatest common factor is 7.

2. **Divide both the numerator and denominator by the GCD (7).**
- \(-\frac{14 \div 7}{49 \div 7} = -\frac{2}{7}\)

**Standard form: \(-\frac{2}{7}\)**

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**(iii) Simplifying \(-\frac{24}{64}\)**

1. **Find the GCD of 24 and 64.**
- The factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- The factors of 64: 1, 2, 4, 8, 16, 32, 64
- The common factors are 1, 2, 4, 8. The greatest common factor is 8.

2. **Divide both the numerator and denominator by the GCD (8).**
- \(-\frac{24 \div 8}{64 \div 8} = -\frac{3}{8}\)

**Standard form: \(-\frac{3}{8}\)**

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**(iv) Simplifying \(-\frac{36}{-63}\)**

1. **Find the GCD of 36 and 63.**
- The factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- The factors of 63: 1, 3, 7, 9, 21, 63
- The common factors are 1, 3, 9. The greatest common factor is 9.

2. **Divide both the numerator and denominator by the GCD (9).**
- \(-\frac{36 \div 9}{-63 \div 9} = \frac{4}{7}\) (Note: The negative signs cancel out)

**Standard form: \(\frac{4}{7}\)**

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### Final Answers:
1. \(-\frac{2}{5}\)
2. \(-\frac{2}{7}\)
3. \(-\frac{3}{8}\)
4. \(\frac{4}{7}\)