Which expression is the greatest common factor of (125t^3 m^5+60t^4 m^4+85t^5 m^2)
don't know how the parentheses are supposed to help. If you mean the GCF of
(125t^3) (m^5+60t^4) (m^4+85t^5) (m^2)
I'd say its just the product, since they have no common factor at all.
If you mean
(125t^3 m^5) (60t^4 m^4) (85t^5 m^2)
then we have
125 t^3 m^5 = 5^3 t^3 m^5
60 t^4 m^4 = 2^2 3 5 t^4 m^4
85 t^5 m^2 = 5 17 t^5 m^2
GCF = 5 t^3 m^2
Look for highest power of each prime/variable that appears in all expressions
To determine the greatest common factor (GCF) of the given expression (125t^3 m^5 + 60t^4 m^4 + 85t^5 m^2), we need to look for the common factors in all the terms of the expression and find their highest power.
Step 1: Identify the terms of the expression.
The given expression has three terms: 125t^3 m^5, 60t^4 m^4, and 85t^5 m^2.
Step 2: Identify the factors of each term.
The factors of the first term, 125t^3 m^5, are 5, 5, 5, t, t, t, m, m, m.
The factors of the second term, 60t^4 m^4, are 2, 2, 3, 5, t, t, t, t, m, m, m, m.
The factors of the third term, 85t^5 m^2, are 5, 17, t, t, t, t, t, m, m.
Step 3: Identify the common factors and their highest power.
In order for a factor to be common, it must appear in each term of the expression. Looking at the factors identified in each term, the common factors are 5, t, and m.
To find the highest power of each common factor, we look for the lowest exponent of that factor in all terms. Here's how we determine the highest power of each common factor:
- 5 appears in the first term with an exponent of 1, in the second term with an exponent of 0, and in the third term with an exponent of 0. Therefore, the highest power of 5 is 0.
- t appears in the first term with an exponent of 3, in the second term with an exponent of 4, and in the third term with an exponent of 5. Therefore, the highest power of t is 3.
- m appears in the first term with an exponent of 5, in the second term with an exponent of 4, and in the third term with an exponent of 2. Therefore, the highest power of m is 2.
Step 4: Write the GCF of the expression.
Now that we have determined the highest power of each common factor, we can write the GCF of the expression: 5^0 * t^3 * m^2, which simplifies to t^3 * m^2.
Therefore, the expression t^3 * m^2 is the greatest common factor (GCF) of the given expression (125t^3 m^5 + 60t^4 m^4 + 85t^5 m^2).