To find the value of (a-b)^4, we can start by expanding the expression using the binomial theorem:
(a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4
Now, we are given that (ab)^3 = 125, so we can rewrite this as:
a^3b^3 = 125
Taking the square root of both sides, we get:
ab = 5
Next, we are given that a^2 + b^2 = 15. We can rewrite this as:
(a^2 + b^2)^2 = 225
Expanding the left side, we get:
a^4 + 2a^2b^2 + b^4 = 225
Now, we know that ab = 5, so we can substitute this into the expression:
a^4 + 2(5^2) + b^4 = 225
a^4 + 50 + b^4 = 225
a^4 + b^4 = 175
Now, we can substitute this back into the expression for (a-b)^4:
(a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4(a-b)^4 = a^4 - 4a^3b + 6(5^2) - 4(5)b + b^4(a-b)^4 = a^4 - 4a^3b + 150 - 20b + b^4(a-b)^4 = a^4 - 4a^3b - 20b + 150
Therefore, (a-b)^4 = a^4 - 4a^3b - 20b + 150
To find the value of (a-b)^4, we can start by expanding the expression using the binomial theorem:
(a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4
Now, we are given that (ab)^3 = 125, so we can rewrite this as:
a^3b^3 = 125
Taking the square root of both sides, we get:
ab = 5
Next, we are given that a^2 + b^2 = 15. We can rewrite this as:
(a^2 + b^2)^2 = 225
Expanding the left side, we get:
a^4 + 2a^2b^2 + b^4 = 225
Now, we know that ab = 5, so we can substitute this into the expression:
a^4 + 2(5^2) + b^4 = 225
a^4 + 50 + b^4 = 225
a^4 + b^4 = 175
Now, we can substitute this back into the expression for (a-b)^4:
(a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4
(a-b)^4 = a^4 - 4a^3b + 6(5^2) - 4(5)b + b^4
(a-b)^4 = a^4 - 4a^3b + 150 - 20b + b^4
(a-b)^4 = a^4 - 4a^3b - 20b + 150
Therefore, (a-b)^4 = a^4 - 4a^3b - 20b + 150