Find the greatest common factor of 30p Superscript 4 Baseline d cubed​, 45p Superscript 6 Baseline d Superscript 5​, and 90p Superscript 9 Baseline d Superscript 4.

Answer 1

To find the greatest common factor (GCF) of the expressions \(30p^4d^3\), \(45p^6d^5\), and \(90p^9d^4\), we will first find the GCF of the numerical coefficients and then the GCF of the variable parts.

Step 1: GCF of the numerical coefficients

The coefficients are 30, 45, and 90.

1. Factors of 30: \(30 = 2 \times 3 \times 5\)
2. Factors of 45: \(45 = 3^2 \times 5\)
3. Factors of 90: \(90 = 2 \times 3^2 \times 5\)

Now, list the common prime factors:

• The common primes are \(3\) and \(5\).
• For \(3\), the lowest power among the coefficients is \(3^1\).
• For \(5\), the lowest power is \(5^1\).

Calculating the GCF: \[ \text{GCF} = 3^1 \times 5^1 = 15 \]

Step 2: GCF of the variable parts

Next, we will find the GCF of the variable parts \(p^4d^3\), \(p^6d^5\), and \(p^9d^4\).

1. For \(p\):

   • We have \(p^4\), \(p^6\), and \(p^9\).
   • The smallest exponent is \(4\).

   Thus, the GCF for \(p\) is \(p^4\).

2. For \(d\):

   • We have \(d^3\), \(d^5\), and \(d^4\).
   • The smallest exponent is \(3\).

   Thus, the GCF for \(d\) is \(d^3\).

Step 3: Combining the results

Now, combine the GCF of the coefficients with the GCF of the variable parts: \[ \text{GCF} = 15p^4d^3 \]

Thus, the greatest common factor of \(30p^4d^3\), \(45p^6d^5\), and \(90p^9d^4\) is \[ \boxed{15p^4d^3} \]