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

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3 answers

To factor out the greatest common factor (GCF) for the numbers 30 and 48, we first need to determine what their GCF is.

  1. Find the GCF of 30 and 48:
    • The prime factorization of 30 is: \[ 30 = 2 \times 3 \times 5 \]
    • The prime factorization of 48 is: \[ 48 = 2^4 \times 3 \]
    • The common prime factors are \(2\) and \(3\).
    • The lowest power of \(2\) in both factorizations is \(2^1\) and for \(3\) it is \(3^1\).

Thus, the GCF is: \[ GCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

  1. Apply the distributive property:

    • We can write both 30 and 48 as a multiple of their GCF: \[ 30 = 6 \times 5 \] \[ 48 = 6 \times 8 \]
    • Now we can express them in a form that shows the GCF: \[ 30 + 48 = 6 \times 5 + 6 \times 8 \]

  2. Use the distributive property to factor out the GCF: \[ 30 + 48 = 6(5 + 8) \]

So, the final expression factoring out the greatest common factor is: \[ 6(5 + 8) \]

This demonstrates the application of the distributive property by factoring out the GCF of 30 and 48.

what is this ^

The symbol ^ is used to denote exponentiation in mathematics. For example, \(2^3\) means \(2\) raised to the power of \(3\), which equals \(2 \times 2 \times 2 = 8\).

In the context of my previous response, I used it to indicate powers of numbers:

  • \(2^4\) means \(2\) raised to the power of \(4\) (which equals \(16\)).
  • \(2^1\) means \(2\) raised to the power of \(1\), which is simply \(2\).
  • \(3^1\) means \(3\) raised to the power of \(1\), which is simply \(3\).

If you have any further questions or need clarification on anything else, feel free to ask!