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To determine which of the given terms can appear in the expansion of [tex]\((2a + 4b)^8\)[/tex], we need to understand the binomial expansion theorem. The theorem states that the general term in the expansion of [tex]\((x + y)^n\)[/tex] is given by:
[tex]\[ \binom{n}{k} x^{n-k} y^k \][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient.
For our specific problem, [tex]\(n = 8\)[/tex], [tex]\(x = 2a\)[/tex], and [tex]\(y = 4b\)[/tex].
The general term in the expansion of [tex]\((2a + 4b)^8\)[/tex] will therefore be of the form:
[tex]\[ \binom{8}{k} \cdot (2a)^{8-k} \cdot (4b)^k \][/tex]
Simplifying, this becomes:
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot a^{8-k} \cdot 4^k \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot 4^k \cdot a^{8-k} \cdot b^k \][/tex]
To combine the powers of 2, we note that [tex]\(4^k = (2^2)^k = 2^{2k}\)[/tex]:
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot 2^{2k} \cdot a^{8-k} \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8-k+2k} \cdot a^{8-k} \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8+k} \cdot a^{8-k} \cdot b^k \][/tex]
We are interested in the exponents of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. For a term [tex]\(a^m b^n\)[/tex] to be in the expansion, [tex]\((8-k)\)[/tex] must equal [tex]\(m\)[/tex] and [tex]\(k\)[/tex] must equal [tex]\(n\)[/tex]. Importantly, [tex]\(k\)[/tex] must be an integer between 0 and 8 (inclusive) because [tex]\(k\)[/tex] ranges from 0 to 8.
Let's check the given terms:
1. [tex]\(a^2 b^3\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(2\)[/tex]: [tex]\(8 - k = 2\)[/tex] gives [tex]\(k = 6\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(k = 3\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
2. [tex]\(a^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(8 - k = 8\)[/tex] gives [tex]\(k = 0\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(0\)[/tex]: [tex]\(k = 0\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
3. [tex]\(a^5 b^3\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(8 - k = 5\)[/tex] gives [tex]\(k = 3\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(k = 3\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
4. [tex]\(a b^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(1\)[/tex]: [tex]\(8 - k = 1\)[/tex] gives [tex]\(k = 7\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(k = 8\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
5. [tex]\(a^3 b^5\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(8 - k = 3\)[/tex] gives [tex]\(k = 5\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(k = 5\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
6. [tex]\(a^7 b\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(7\)[/tex]: [tex]\(8 - k = 7\)[/tex] gives [tex]\(k = 1\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(1\)[/tex]: [tex]\(k = 1\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
7. [tex]\(a^6 b^5\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(6\)[/tex]: [tex]\(8 - k = 6\)[/tex] gives [tex]\(k = 2\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(k = 5\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
8. [tex]\(b^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(0\)[/tex]: [tex]\(8 - k = 0\)[/tex] gives [tex]\(k = 8\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(k = 8\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
The possible terms in the expansion of [tex]\((2a + 4b)^8\)[/tex] are:
[tex]\[ a^8, a^5 b^3, a^3 b^5, a^7 b, b^8 \][/tex]
[tex]\[ \binom{n}{k} x^{n-k} y^k \][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient.
For our specific problem, [tex]\(n = 8\)[/tex], [tex]\(x = 2a\)[/tex], and [tex]\(y = 4b\)[/tex].
The general term in the expansion of [tex]\((2a + 4b)^8\)[/tex] will therefore be of the form:
[tex]\[ \binom{8}{k} \cdot (2a)^{8-k} \cdot (4b)^k \][/tex]
Simplifying, this becomes:
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot a^{8-k} \cdot 4^k \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot 4^k \cdot a^{8-k} \cdot b^k \][/tex]
To combine the powers of 2, we note that [tex]\(4^k = (2^2)^k = 2^{2k}\)[/tex]:
[tex]\[ \binom{8}{k} \cdot 2^{8-k} \cdot 2^{2k} \cdot a^{8-k} \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8-k+2k} \cdot a^{8-k} \cdot b^k \][/tex]
[tex]\[ \binom{8}{k} \cdot 2^{8+k} \cdot a^{8-k} \cdot b^k \][/tex]
We are interested in the exponents of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. For a term [tex]\(a^m b^n\)[/tex] to be in the expansion, [tex]\((8-k)\)[/tex] must equal [tex]\(m\)[/tex] and [tex]\(k\)[/tex] must equal [tex]\(n\)[/tex]. Importantly, [tex]\(k\)[/tex] must be an integer between 0 and 8 (inclusive) because [tex]\(k\)[/tex] ranges from 0 to 8.
Let's check the given terms:
1. [tex]\(a^2 b^3\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(2\)[/tex]: [tex]\(8 - k = 2\)[/tex] gives [tex]\(k = 6\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(k = 3\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
2. [tex]\(a^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(8 - k = 8\)[/tex] gives [tex]\(k = 0\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(0\)[/tex]: [tex]\(k = 0\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
3. [tex]\(a^5 b^3\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(8 - k = 5\)[/tex] gives [tex]\(k = 3\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(k = 3\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
4. [tex]\(a b^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(1\)[/tex]: [tex]\(8 - k = 1\)[/tex] gives [tex]\(k = 7\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(k = 8\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
5. [tex]\(a^3 b^5\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(3\)[/tex]: [tex]\(8 - k = 3\)[/tex] gives [tex]\(k = 5\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(k = 5\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
6. [tex]\(a^7 b\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(7\)[/tex]: [tex]\(8 - k = 7\)[/tex] gives [tex]\(k = 1\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(1\)[/tex]: [tex]\(k = 1\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
7. [tex]\(a^6 b^5\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(6\)[/tex]: [tex]\(8 - k = 6\)[/tex] gives [tex]\(k = 2\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(5\)[/tex]: [tex]\(k = 5\)[/tex].
- [tex]\(k\)[/tex] values do not match, so this term is not possible.
8. [tex]\(b^8\)[/tex]:
- [tex]\(a\)[/tex] has exponent [tex]\(0\)[/tex]: [tex]\(8 - k = 0\)[/tex] gives [tex]\(k = 8\)[/tex].
- [tex]\(b\)[/tex] has exponent [tex]\(8\)[/tex]: [tex]\(k = 8\)[/tex].
- The [tex]\(k\)[/tex] values match, so this term is possible.
The possible terms in the expansion of [tex]\((2a + 4b)^8\)[/tex] are:
[tex]\[ a^8, a^5 b^3, a^3 b^5, a^7 b, b^8 \][/tex]
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