In the expression below if a = 4, what value of b is needed for the expression to have a value of 81?
12a^3b^7(ab)^-4
To find the value of b that would make the expression have a value of 81, we can substitute the given value of a = 4 into the expression:
12(4)^3b^7(4b)^-4
Simplifying further:
12(64)b^7(4b)^-4
768b^7(4b)^-4
To find the value of b that would make the expression equal to 81, we set it equal to 81 and solve for b:
768b^7(4b)^-4 = 81
Dividing both sides by 768 to isolate b:
b^7(4b)^-4 = 81/768
Taking the 7th root of both sides:
b(4b)^-4/7 = (81/768)^(1/7)
Simplifying the right side:
b(4b)^-4/7 = (1/9)
Multiplying both sides by (4b)^4/7:
b = (4b)^4/7 / 9
Now we can solve for b using an iterative process or a calculator. Let's use a calculator to find the approximate value of b:
b ≈ 3.315
Therefore, the value of b needed for the expression to have a value of 81 when a = 4 is approximately 3.315.