Question

Use the binomial theorem to expand (a b)6.

Answer 1

To expand (a b)^6 using the binomial theorem, we can utilize the formula (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n. Applying this formula, we can expand (a b)^6 as a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6.

Step-by-step explanation:

The binomial theorem is an algebraic formula used to expand binomial expressions. In this case, we need to expand (a b)^6. The binomial theorem states that (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n, where nCk is the binomial coefficient.

For (a b)^6, the exponent is 6. Applying the formula, we have:

1. Coefficient for the first term: 6C0 = 1
2. a^6 * b^0 = a^6
3. Coefficient for the second term: 6C1 = 6
4. a^5 * b^1 = 6a^5b
5. Coefficient for the third term: 6C2 = 15
6. a^4 * b^2 = 15a^4b^2
7. Coefficient for the fourth term: 6C3 = 20
8. a^3 * b^3 = 20a^3b^3
9. Coefficient for the fifth term: 6C4 = 15
10. a^2 * b^4 = 15a^2b^4
11. Coefficient for the sixth term: 6C5 = 6
12. a^1 * b^5 = 6ab^5
13. Coefficient for the seventh term: 6C6 = 1
14. a^0 * b^6 = b^6

Combining all the terms, we get (a b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6.