In the expansion of (2a + 4b)8, which of the following are possible variable terms? Explain your reasoning.
a2b3; a8; a5b3; ab8; a3b5; a7b; a6b5; b8
To find the variable terms in the expansion of (2a + 4b)^8, we need to consider the terms that involve both 'a' and 'b'.
Looking at the possible terms:
a^2b^3 - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
a^8 - This is not a valid term because 'b' does not appear in it.
a^5b^3 - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
ab^8 - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
a^3b^5 - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
a^7b - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
a^6b^5 - This is a valid term since it involves both 'a' and 'b' raised to powers greater than zero.
b^8 - This is not a valid term because 'a' does not appear in it.
Therefore, the possible variable terms in the expansion of (2a + 4b)^8 are: a^2b^3, a^5b^3, ab^8, a^3b^5, a^7b, and a^6b^5.
To determine whether a term is possible in the expansion of (2a + 4b)^8, we need to consider the variables, their exponents, and the coefficients involved.
In the expansion, each term is obtained by choosing either 2a or 4b from each of the eight factors and multiplying them together.
Let's go through each option to see which ones are possible:
a2b3:
To obtain a2b3, we would need to choose the term 2a twice and the term 4b three times. However, since we only have eight factors to choose from, it is not possible to choose 4b three times. Therefore, a2b3 is not a possible term.
a8:
Similarly, to obtain a8, we would need to choose the term 2a eight times. However, we only have eight factors to choose from, so it is not possible to choose 2a eight times. Therefore, a8 is not a possible term.
a5b3:
To obtain a5b3, we would need to choose the term 2a five times and the term 4b three times. Since we have eight factors to choose from, it is possible to choose the terms accordingly. Therefore, a5b3 is a possible term.
ab8:
To obtain ab8, we would need to choose the term 2a once and the term 4b eight times. Since we have eight factors to choose from, it is possible to choose the terms accordingly. Therefore, ab8 is a possible term.
a3b5:
To obtain a3b5, we would need to choose the term 2a three times and the term 4b five times. Since we have eight factors to choose from, it is possible to choose the terms accordingly. Therefore, a3b5 is a possible term.
a7b:
To obtain a7b, we would need to choose the term 2a seven times and the term 4b once. However, we only have eight factors to choose from, so it is not possible to choose 2a seven times. Therefore, a7b is not a possible term.
a6b5:
To obtain a6b5, we would need to choose the term 2a six times and the term 4b five times. Since we have eight factors to choose from, it is possible to choose the terms accordingly. Therefore, a6b5 is a possible term.
b8:
To obtain b8, we would need to choose the term 4b eight times. Since we have eight factors to choose from, it is possible to choose the terms accordingly. Therefore, b8 is a possible term.
In summary, the possible variable terms in the expansion of (2a + 4b)^8 are: a5b3, ab8, a3b5, a6b5, and b8.