Write the following fractions in simplest form:
`168/(420)`

Answer

To simplify the fraction \( \frac{168}{420} \), we will follow these steps:

### Step 1: Find the Greatest Common Divisor (GCD)
We need to find the GCD of 168 and 420. We can do this by listing the factors or using prime factorization.

**Hint:** To find the GCD, you can use the method of prime factorization or list the factors of both numbers.

### Step 2: Prime Factorization
Let's find the prime factorization of both numbers.

- **For 168:**
- Divide by 2: \( 168 \div 2 = 84 \)
- Divide by 2: \( 84 \div 2 = 42 \)
- Divide by 2: \( 42 \div 2 = 21 \)
- Divide by 3: \( 21 \div 3 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 168 is:
\[ 168 = 2^3 \times 3^1 \times 7^1 \]

- **For 420:**
- Divide by 2: \( 420 \div 2 = 210 \)
- Divide by 2: \( 210 \div 2 = 105 \)
- Divide by 3: \( 105 \div 3 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

### Step 3: Identify the Common Factors
Now, we can identify the common prime factors:
- The common factors are \( 2^2 \), \( 3^1 \), and \( 7^1 \).

### Step 4: Calculate the GCD
To find the GCD, we take the lowest power of each common prime factor:
\[ GCD = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84 \]

**Hint:** The GCD is the product of the lowest powers of all common prime factors.

### Step 5: Simplify the Fraction
Now we divide both the numerator and the denominator by the GCD (84):
\[ \frac{168 \div 84}{420 \div 84} = \frac{2}{5} \]

### Final Answer
Thus, the simplest form of the fraction \( \frac{168}{420} \) is:
\[ \frac{2}{5} \]