Question

find the greatest common factor.
3q^{3}, 3q^{2}
write your answer as a constant times a product of single variables raised to exponents.

Explanation:

Step1: Analyze coefficients

The coefficients are 3 (for both terms). The GCF of 3 and 3 is 3.

Step2: Analyze variables

For the variable \( q \), the first term has \( q^3 \) and the second has \( q^2 \). The lowest power of \( q \) is \( q^2 \)? Wait, no, wait. Wait, the terms are \( 3q^3 \) and \( 3q \)? Wait, no, the original problem: let me check again. Wait, the user's image: "3q³, 3q²"? Wait, no, the user's text: "3q ³, 3q ²"? Wait, no, the OCR: "3q ³, 3q ²" (maybe 3q³ and 3q²? Or 3q³ and 3q? Wait, the user's image: "3q ³, 3q ²" – maybe 3q³ and 3q²? Wait, no, the original problem: "Find the greatest common factor. 3q³, 3q². Write your answer as a constant times a product of single variables raised to exponents." Wait, no, maybe 3q³ and 3q? Wait, the OCR: "3q ³, 3q ²" – perhaps 3q³ and 3q²? Wait, no, let's re-express. Let's take the two terms: \( 3q^3 \) and \( 3q^2 \)? Wait, no, the user's image: "3q ³, 3q ²" – maybe 3q³ and 3q²? Wait, no, maybe 3q³ and 3q? Wait, the OCR is a bit unclear. Wait, the problem says "3q ³, 3q ²" – perhaps 3q³ and 3q²? Wait, no, let's check the exponents. For the variable part: the first term is \( q^3 \), the second is \( q^2 \)? No, wait, maybe the terms are \( 3q^3 \) and \( 3q \)? Wait, no, the user's text: "3q ³, 3q ²" – maybe 3q³ and 3q². Wait, no, let's do it properly. Let's take the two terms: \( 3q^3 \) and \( 3q^2 \). Wait, no, maybe the terms are \( 3q^3 \) and \( 3q \). Wait, the OCR is "3q ³, 3q ²" – maybe 3q³ and 3q². Wait, no, let's re-express. Let's suppose the terms are \( 3q^3 \) and \( 3q^2 \). Then, for the variable \( q \), the exponents are 3 and 2. The GCF for the variable is \( q^2 \)? No, wait, no: the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, the GCF of \( q^3 \) (which is \( q \times q \times q \)) and \( q^2 \) (which is \( q \times q \)) is \( q^2 \)? Wait, no, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, the GCF of \( q^3 \) (factors: \( q \times q \times q \)) and \( q^2 \) (factors: \( q \times q \)) is \( q \times q = q^2 \)? Wait, no, the GCF is the product of the lowest powers of all common prime factors. So for the variable \( q \), the lowest power is \( q^2 \)? Wait, no, wait: if the terms are \( 3q^3 \) and \( 3q^2 \), then the GCF of the coefficients is 3, and the GCF of the variables is \( q^2 \), so the overall GCF is \( 3 \times q^2 = 3q^2 \)? Wait, no, that can't be. Wait, no, maybe the terms are \( 3q^3 \) and \( 3q \). Wait, the OCR is "3q ³, 3q ²" – maybe 3q³ and 3q². Wait, let's start over. Let's take the two terms: \( 3q^3 \) and \( 3q^2 \). Step 1: Coefficients: 3 and 3. GCF of 3 and 3 is 3. Step 2: Variables: \( q^3 \) and \( q^2 \). The GCF of \( q^3 \) (which is \( q \times q \times q \)) and \( q^2 \) (which is \( q \times q \)) is \( q^2 \)? No, wait, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, the GCF is the product of the minimum exponents. So for \( q \), the exponents are 3 and 2. The minimum exponent is 2? Wait, no, 2 is less than 3, so the GCF for \( q \) is \( q^2 \)? Wait, no, that's not right. Wait, no, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \)? Wait, no, let's take an example. Let \( q = 2 \). Then \( q^3 = 8 \), \( q^2 = 4 \). The GCF of 8 and 4 is 4, which is \( 2^2 \), which is \( q^2 \) when \( q=2 \). So yes, the GCF of \( q^3 \) and \( q^2 \) is \( q^2 \). Wait, but then the GCF of \( 3q^3 \) and \( 3q^2 \) would be \( 3 \times q^2 = 3q^2 \). But wait, maybe the terms are \( 3q^3 \) and \( 3q \). Let's check. If the terms are \( 3q^3 \) and \( 3q \), then the coefficient GCF is 3, and the variable GCF is \( q \) (since \( q^3 \) and \( q \) have GCF \( q \)). Then the GCF would be \( 3q \). Wait, the user's image: "3q ³, 3q ²" – maybe 3q³ and 3q². Wait, the OCR is "3q ³, 3q ²" – perhaps 3q³ and 3q². Let's confirm. Let's factor both terms:

- \( 3q^3 = 3 \times q \times q \times q \)
- \( 3q^2 = 3 \times q \times q \)

The common factors are 3, \( q \), \( q \). So the GCF is \( 3 \times q \times q = 3q^2 \)? Wait, no, \( 3q^3 = 3 \times q \times q \times q \), \( 3q^2 = 3 \times q \times q \). The common factors are 3, \( q \), \( q \), so GCF is \( 3q^2 \). But wait, maybe the terms are \( 3q^3 \) and \( 3q \). Let's see:

- \( 3q^3 = 3 \times q \times q \times q \)
- \( 3q = 3 \times q \)

Common factors: 3, \( q \). So GCF is \( 3q \).

Wait, the user's image: "3q ³, 3q ²" – maybe 3q³ and 3q². Let's check the original problem again. The user's image: "Find the greatest common factor. 3q ³, 3q ². Write your answer as a constant times a product of single variables raised to exponents." So the two terms are \( 3q^3 \) and \( 3q^2 \). Then:

Step 1: Coefficients: 3 and 3. GCF of 3 and 3 is 3.

Step 2: Variables: \( q^3 \) and \( q^2 \). The GCF of \( q^3 \) and \( q^2 \) is \( q^2 \) (since the lowest power of \( q \) is 2? Wait, no, \( q^3 \) is \( q \times q \times q \), \( q^2 \) is \( q \times q \). The common factors for the variables are \( q \times q \), so \( q^2 \).

Wait, but that would make the GCF \( 3 \times q^2 = 3q^2 \). But let's verify with numbers. Let \( q = 2 \). Then \( 3q^3 = 3 \times 8 = 24 \), \( 3q^2 = 3 \times 4 = 12 \). The GCF of 24 and 12 is 12, which is \( 3 \times 4 = 3 \times 2^2 = 3q^2 \) (since \( q=2 \)). So that works.

But wait, maybe the terms are \( 3q^3 \) and \( 3q \). Let's check. If \( q=2 \), \( 3q^3=24 \), \( 3q=6 \). GCF of 24 and 6 is 6, which is \( 3 \times 2 = 3q \). So that also works.

Wait, the user's image: "3q ³, 3q ²" – the OCR is "3q ³, 3q ²" – so 3q³ and 3q². So the two terms are \( 3q^3 \) and \( 3q^2 \). Then the GCF is \( 3q^2 \)? Wait, no, wait: \( 3q^3 = 3 \times q \times q \times q \), \( 3q^2 = 3 \times q \times q \). The common factors are 3, q, q. So GCF is \( 3 \times q \times q = 3q^2 \). But wait, let's check with another example. Let \( q = 5 \). Then \( 3q^3 = 3 \times 125 = 375 \), \( 3q^2 = 3 \times 25 = 75 \). The GCF of 375 and 75 is 75, which is \( 3 \times 25 = 3 \times 5^2 = 3q^2 \) (since \( q=5 \)). So that's correct.

Wait, but maybe the user made a typo, and the terms are \( 3q^3 \) and \( 3q \). Let's see. If the terms are \( 3q^3 \) and \( 3q \), then GCF is \( 3q \). But according to the OCR, it's "3q ³, 3q ²" – so 3q³ and 3q². So proceeding with that.

So step 1: GCF of coefficients (3 and 3) is 3.

Step 2: GCF of variables (q³ and q²) is q² (since the lowest exponent is 2? Wait, no, the exponents are 3 and 2. The GCF of q³ and q² is q² because q² is a factor of q³ (q³ = q² × q). So yes, q² is the GCF of the variable parts.

Therefore, the GCF of 3q³ and 3q² is 3 × q² = 3q².

Wait, but let's check again. The GCF of 3q³ and 3q²:

Prime factors of 3q³: 3, q, q, q.

Prime factors of 3q²: 3, q, q.

Common prime factors: 3, q, q.

Multiply them: 3 × q × q = 3q².

Yes, that's correct.

Answer:

\( 3q^2 \)