Final answer:
To expand the binomial (8v + s)^5, use the coefficients from Pascal's Triangle and apply the binomial expansion formula, resulting in 32768v^5 + 20480v^4s + 5120v^3s^2 + 640v^2s^3 + 40vs^4 + s^5.
Step-by-step explanation:
To expand a binomial using Pascal's Triangle, we apply the coefficients found in the row of Pascal's Triangle that corresponds to the power of the binomial expansion. For the expansion of (8v + s)^5, we look at the fifth row (starting with row 0), which provides the coefficients 1, 5, 10, 10, 5, 1. The binomial expansion can be expressed as:
a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
By substituting a with 8v and b with s, we get:
(8v)^5 + 5(8v)^4s + 10(8v)^3s^2 + 10(8v)^2s^3 + 5(8v)s^4 + s^5
After calculating the powers of 8v and multiplying by the corresponding coefficients, we get the expanded form:
32768v^5 + 5(4096)v^4s + 10(512)v^3s^2 + 10(64)v^2s^3 + 5(8)vs^4 + s^5
Which simplifies to:
32768v^5 + 20480v^4s + 5120v^3s^2 + 640v^2s^3 + 40vs^4 + s^5