Question

find the greatest common factor.
7, 7a²b⁸c⁵, 49a³b¹⁰c²
write your answer as a constant times a product of single variables raised to exponents.

Explanation:

Step1: Find GCF of coefficients

The coefficients are 7, 7, and 49. The GCF of 7, 7, 49 is 7.

Step2: Find GCF of \(a\) terms

For \(a\) terms: \(a^0\) (from 7), \(a^2\) (from \(7a^2b^8c^5\)), \(a^3\) (from \(49a^3b^{10}c^2\)). The lowest power of \(a\) is \(a^0 = 1\), but wait, no—wait, the first term is 7 (which is \(7a^0b^0c^0\)), the second is \(7a^2b^8c^5\), the third is \(49a^3b^{10}c^2\). So for \(a\): exponents are 0, 2, 3. The GCF for \(a\) is \(a^0 = 1\)? Wait, no, maybe I misread. Wait the first term is 7, then 7\(a^2\)b\(^8\)c\(^5\), then 49\(a^3\)b\(^{10}\)c\(^2\). So for \(a\): the exponents are 0 (in 7), 2 (in \(7a^2b^8c^5\)), 3 (in \(49a^3b^{10}c^2\)). The greatest common factor for the \(a\) part is \(a^0 = 1\) (since 0 is the lowest exponent). Wait, no, maybe the first term is 7 (no \(a\)), the second has \(a^2\), the third has \(a^3\). So the GCF for \(a\) is \(a^0\) (since 0 is the minimum exponent among 0, 2, 3). But wait, maybe the first term is 7 (with \(a^0\)), so the GCF for \(a\) is \(a^0 = 1\), but actually, maybe I made a mistake. Wait, no—wait, the first term is 7, which is \(7 \times 1 \times 1 \times 1\) (in terms of \(a\), \(b\), \(c\)). The second term is \(7 \times a^2 \times b^8 \times c^5\). The third term is \(49 \times a^3 \times b^{10} \times c^2\). So for \(a\): the exponents are 0, 2, 3. The GCF is \(a^0 = 1\) (since 0 is the smallest). For \(b\): exponents are 0 (in 7), 8 (in \(7a^2b^8c^5\)), 10 (in \(49a^3b^{10}c^2\)). So GCF for \(b\) is \(b^0 = 1\)? Wait, no, that can't be. Wait, no, the first term is 7 (no \(b\)), so \(b^0\), the second has \(b^8\), the third has \(b^{10}\). So the GCF for \(b\) is \(b^0 = 1\)? Wait, that seems wrong. Wait, maybe the first term is 7 (with \(b^0\)), the second has \(b^8\), the third has \(b^{10}\). So the GCF for \(b\) is \(b^0 = 1\). For \(c\): exponents are 0 (in 7), 5 (in \(7a^2b^8c^5\)), 2 (in \(49a^3b^{10}c^2\)). So GCF for \(c\) is \(c^0 = 1\). Wait, but that would make the GCF 7 \(\times\) 1 \(\times\) 1 \(\times\) 1 = 7. But that seems too simple. Wait, maybe I misread the first term. Wait the first term is 7, then 7\(a^2\)b\(^8\)c\(^5\), then 49\(a^3\)b\(^{10}\)c\(^2\). Wait, maybe the first term is 7 (with \(a^0\), \(b^0\), \(c^0\)), the second is \(7a^2b^8c^5\), the third is \(49a^3b^{10}c^2\). So let's re-express each term:

- Term 1: \(7 = 7 \times a^0 \times b^0 \times c^0\)
- Term 2: \(7a^2b^8c^5 = 7 \times a^2 \times b^8 \times c^5\)
- Term 3: \(49a^3b^{10}c^2 = 7 \times 7 \times a^3 \times b^{10} \times c^2\)

Now, find GCF for each part:

- Coefficients: GCF of 7, 7, 49 is 7.
- \(a\) terms: exponents 0, 2, 3. GCF exponent is 0, so \(a^0 = 1\).
- \(b\) terms: exponents 0, 8, 10. GCF exponent is 0, so \(b^0 = 1\).
- \(c\) terms: exponents 0, 5, 2. GCF exponent is 0, so \(c^0 = 1\).

Wait, that can't be right. Wait, maybe the first term is 7 (with no variables), the second has \(a^2\), \(b^8\), \(c^5\), the third has \(a^3\), \(b^{10}\), \(c^2\). So the only common factors are the coefficient 7 and no variables? But that seems odd. Wait, maybe I misread the first term. Wait, the problem says "7, 7\(a^2\)b\(^8\)c\(^5\), 49\(a^3\)b\(^{10}\)c\(^2\)". So the three terms are 7, \(7a^2b^8c^5\), and \(49a^3b^{10}c^2\). So to find the GCF, we look for the greatest factor that divides all three.

For the coefficient: 7 divides 7, 7, and 49 (since 49 ÷ 7 = 7). So coefficient GCF is 7.

For the \(a\) variable: The first term has \(a^0\) (no \(a\)), the second has \(a^2\), the third has \(a^3\). The highest power of \(a\) that divides all three is \(a^0\) (since \(a^0 = 1\) divides \(a^2\) and \(a^3\), and is in the first term).

For the \(b\) variable: First term has \(b^0\), second has \(b^8\), third has \(b^{10}\). Highest power of \(b\) that divides all three is \(b^0\).

For the \(c\) variable: First term has \(c^0\), second has \(c^5\), third has \(c^2\). Highest power of \(c\) that divides all three is \(c^0\).

So putting it together, the GCF is \(7 \times a^0 \times b^0 \times c^0 = 7 \times 1 \times 1 \times 1 = 7\). Wait, but that seems too simple. Wait, maybe the first term is 7 (with no variables), the second is \(7a^2b^8c^5\), the third is \(49a^3b^{10}c^2\). So the GCF is 7, because 7 is a factor of all three terms (7 ÷ 7 = 1, \(7a^2b^8c^5\) ÷ 7 = \(a^2b^8c^5\), \(49a^3b^{10}c^2\) ÷ 7 = \(7a^3b^{10}c^2\)). And there are no common variables because the first term has no variables. So the GCF is 7.

Wait, but maybe I made a mistake. Let's check again. The three terms are 7, \(7a^2b^8c^5\), and \(49a^3b^{10}c^2\). So:

- 7 is a factor of 7 (7 ÷ 7 = 1)
- 7 is a factor of \(7a^2b^8c^5\) (7 ÷ 7 = 1, so \(7a^2b^8c^5\) ÷ 7 = \(a^2b^8c^5\))
- 7 is a factor of \(49a^3b^{10}c^2\) (49 ÷ 7 = 7, so \(49a^3b^{10}c^2\) ÷ 7 = \(7a^3b^{10}c^2\))

Now, for the variables:

- The first term has no \(a\), \(b\), or \(c\), so the other terms have \(a\), \(b\), \(c\) but the first term doesn't. So there's no common variable factor (since the first term has none). Therefore, the GCF is just 7.

Answer:

\(7\)