Express the fractions in their simplest forms:
`24/(42)`

Answer

To express the fraction \( \frac{24}{42} \) in its simplest form, we will follow these steps:

### Step 1: Factorize the Numerator and Denominator
First, we need to factorize both 24 and 42 into their prime factors.

**Factorization of 24:**
- 24 is an even number, so we can divide it by 2:
\[ 24 \div 2 = 12 \]
- 12 is also even, divide it by 2 again:
\[ 12 \div 2 = 6 \]
- 6 is again even, divide it by 2:
\[ 6 \div 2 = 3 \]
- Finally, 3 is a prime number, and dividing it by itself gives:
\[ 3 \div 3 = 1 \]
Thus, the prime factorization of 24 is:
\[ 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \]

**Factorization of 42:**
- 42 is also even, so we divide it by 2:
\[ 42 \div 2 = 21 \]
- 21 is not even, but it can be divided by 3:
\[ 21 \div 3 = 7 \]
- 7 is a prime number, and dividing it by itself gives:
\[ 7 \div 7 = 1 \]
Thus, the prime factorization of 42 is:
\[ 42 = 2 \times 3 \times 7 \]

### Step 2: Rewrite the Fraction with Prime Factors
Now we can rewrite the fraction \( \frac{24}{42} \) using the prime factorization:
\[ \frac{24}{42} = \frac{2^3 \times 3}{2 \times 3 \times 7} \]

### Step 3: Cancel Common Factors
Next, we will cancel the common factors in the numerator and denominator:
- The factor \( 2 \) appears once in the denominator and three times in the numerator, so we can cancel one \( 2 \).
- The factor \( 3 \) appears once in both the numerator and denominator, so we can cancel \( 3 \).

After canceling, we have:
\[ \frac{2^2}{7} = \frac{4}{7} \]

### Step 4: Conclusion
Thus, the simplest form of the fraction \( \frac{24}{42} \) is:
\[ \frac{4}{7} \]

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