Apply distributive property to factor out the greatest common factor for the numbers 30 and 48. show your work please..

3 answers

To factor out the greatest common factor (GCF) using the distributive property for the numbers 30 and 48, we first need to determine the GCF of these two numbers.

Step 1: Find the GCF of 30 and 48.

We start by finding the prime factorization of each number.

  • The prime factorization of 30 is: \[ 30 = 2 \times 3 \times 5 \]

  • The prime factorization of 48 is: \[ 48 = 2^4 \times 3 \]

Step 2: Identify common factors.

Now we identify the common prime factors:

  • Both 30 and 48 include the factors \(2\) and \(3\).

Step 3: Determine the smallest exponent for each common factor.

  • For \(2\): The smallest power in both factorizations is \(2^1\).
  • For \(3\): The smallest power in both factorizations is \(3^1\).

Step 4: Calculate the GCF.

Now we multiply these common factors together to find the GCF:

\[ \text{GCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

Step 5: Factor out the GCF using the distributive property.

Now we can express both numbers (30 and 48) as their GCF multiplied by another factor:

  • For 30: \[ 30 = 6 \times 5 \]

  • For 48: \[ 48 = 6 \times 8 \]

Final Expression:

Thus, we can use the distributive property to express the relationship between these two numbers as follows:

\[ 30 + 48 = 6 \times 5 + 6 \times 8 \]

Using the distributive property, we can factor out the GCF:

\[ 30 + 48 = 6 \times (5 + 8) \]

Calculating the expression inside the parentheses:

\[ 5 + 8 = 13 \]

So, we have:

\[ 30 + 48 = 6 \times 13 \]

In summary, the greatest common factor of 30 and 48 is 6, and we can factor the sum \(30 + 48\) as:

\[ 30 + 48 = 6 \times 13 \]

can you give me another example of problems like this?

Sure! Let's work through another example. We will find the greatest common factor (GCF) and apply the distributive property for the numbers 42 and 56.

Step 1: Find the GCF of 42 and 56.

First, we will find the prime factorization of each number.

  • The prime factorization of 42 is: \[ 42 = 2 \times 3 \times 7 \]

  • The prime factorization of 56 is: \[ 56 = 2^3 \times 7 \]

Step 2: Identify common factors.

Now, let’s identify the common prime factors:

  • Both 42 and 56 have the factors \(2\) and \(7\).

Step 3: Determine the smallest exponent for each common factor.

  • For \(2\): The smallest power is \(2^1\).
  • For \(7\): The smallest power is \(7^1\).

Step 4: Calculate the GCF.

Now we multiply these common factors together to find the GCF:

\[ \text{GCF} = 2^1 \times 7^1 = 2 \times 7 = 14 \]

Step 5: Factor out the GCF using the distributive property.

Now let's express both numbers (42 and 56) as their GCF multiplied by another factor:

  • For 42: \[ 42 = 14 \times 3 \]

  • For 56: \[ 56 = 14 \times 4 \]

Final Expression:

Now, we can use the distributive property to express the relationship between these two numbers as follows:

\[ 42 + 56 = 14 \times 3 + 14 \times 4 \]

Using the distributive property, we can factor out the GCF:

\[ 42 + 56 = 14 \times (3 + 4) \]

Calculating the expression inside the parentheses:

\[ 3 + 4 = 7 \]

So, we have:

\[ 42 + 56 = 14 \times 7 \]

In summary, the greatest common factor of 42 and 56 is 14, and we can factor the sum \(42 + 56\) as:

\[ 42 + 56 = 14 \times 7 \]

Additional Practice Problems:

  1. Find the GCF and use the distributive property for the numbers 18 and 24.

  2. Find the GCF and use the distributive property for the numbers 60 and 84.

  3. Find the GCF and use the distributive property for the numbers 81 and 36.

Let me know if you would like me to solve any of these additional problems!