To factor out the greatest common factor (GCF) of the numbers 30 and 48 using the distributive property, we first need to determine what the GCF is.
Step 1: Find the GCF of 30 and 48.
Step 1.1: Find the prime factorization of each number.
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For 30:
- \(30 = 2 \times 3 \times 5\)
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For 48:
- \(48 = 2^4 \times 3\)
Step 1.2: Identify the common factors.
- The common prime factors between 30 and 48 are \(2\) and \(3\).
Step 1.3: Determine the lowest power of each common factor.
- For \(2\): The minimum power is \(2^1\) (from 30).
- For \(3\): The minimum power is \(3^1\) (common in both).
Step 1.4: Calculate the GCF. The GCF is calculated as: \[ GCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]
Step 2: Factor out the GCF from both numbers.
Now that we know the GCF is 6, we can express each of the original numbers (30 and 48) as a product involving this GCF:
- For 30: \[ 30 = 6 \times 5 \]
- For 48: \[ 48 = 6 \times 8 \]
Step 3: Use the distributive property.
We can now combine the factors using the distributive property: \[ 30 + 48 = (6 \times 5) + (6 \times 8) \]
Using the distributive property, we factor out the GCF: \[ 30 + 48 = 6(5 + 8) \]
Final Result:
Thus, by applying the distributive property to factor out the greatest common factor, we have: \[ 30 + 48 = 6(5 + 8) = 6 \times 13 \] This shows how you can factor out the GCF of 30 and 48 using the distributive property.
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