Question

Without using xa*xb=xa+b, show that for any whole number n and any positive number y, (y-1)ⁿ = yⁿ.

A. True
B. False
C. Depends on the values of n and y
D. Insufficient information

Answer 1

To show that (y-1)ⁿ = yⁿ for any whole number n and any positive number y, we can expand both sides using the binomial theorem. By substituting y-1 for a and 1 for b, we can simplify the expression and prove that (y-1)ⁿ = yⁿ.

Step-by-step explanation:

To show that (y-1)ⁿ = yⁿ, we can expand both sides using the binomial theorem. The binomial theorem states that for any positive number y and any whole number n, (a + b)ⁿ = aⁿ + nan⁻¹b + n(n-1)aⁿ⁻²b²/2! + ...

When we substitute y-1 for a and 1 for b in the binomial theorem, we get (y-1 + 1)ⁿ = (y-1)ⁿ + n(y-1)ⁿ⁻¹ + n(n-1)(y-1)ⁿ⁻²/2! + ...

By simplifying, we find that (y-1 + 1)ⁿ = (y-1)ⁿ + n(y-1)ⁿ⁻¹ + n(n-1)(y-1)ⁿ⁻²/2! + ... is equal to yⁿ. Therefore, (y-1)ⁿ = yⁿ is always true, regardless of the values of n and y.