simplify the compound fraction: ((x+h)^(-3)-x^(-3))/(h)
where do i start?
[1/(x+h)^3 - 1/x^3 ]/h
yes
so continue
[ x^3 -(x+h)^3 ] / [h x^3(x+h)^3]
idek the next step damon
(x+h)^2 = x^2 + 2 h x + h^2
x(x^2 + 2 h x + h^2)=x^3 +2 hx^2+ xh^2
h(x^2 + 2 h x + h^2)=hx^2+2h^2x+h^3
sum=x^3 +3hx^2 +3h^2x+ h^3
etc
thats the answer damon sum=x^3 +3hx^2 +3h^2x+ h^3
To simplify the compound fraction ((x+h)^(-3) - x^(-3))/h, you can follow these steps:
Step 1: Simplify the numerator.
Start by applying the binomial formula to both terms in the numerator. The binomial formula states that for any two terms a and b, (a + b)^n = a^n + nC1 * a^n-1 * b^1 + nC2 * a^n-2 * b^2 + ... + nCn-1 * a * b^n-1 + b^n, where nCk represents the binomial coefficient. In this case, we have two terms: (x + h)^(-3) and x^(-3).
Using the binomial formula for (x + h)^(-3), we get:
(x + h)^(-3) = x^(-3) + (-3) * x^(-3-1) * h^1 + (-3)*(-4)/2 * x^(-3-2) * h^2 + ...
Similarly, using the binomial formula for x^(-3), we get:
x^(-3) = 1/x^3
Now, substitute these values back into the original expression:
((x+h)^(-3) - x^(-3)) = (x^(-3) + (-3) * x^(-3-1) * h^1 + (-3)*(-4)/2 * x^(-3-2) * h^2 + ...) - (1/x^3)
Step 2: Simplify the denominator.
The denominator, h, cannot be simplified further.
Step 3: Combine like terms.
In the numerator, we have both x^(-3) and 1/x^3. To combine them, we need to find a common denominator. The common denominator is x^3.
Multiply the first term, x^(-3), by x^(-3)/x^(-3) to get x^(-3) * x^(-3)/x^(-3) = x^(-6)/x^(-3) = 1/x^3.
Now, substitute this value back into the expression:
((x+h)^(-3) - x^(-3))/h = (1/x^3 + (-3) * x^(-4) * h + (-3)*(-4)/2 * x^(-5) * h^2 + ...) - (1/x^3) / h
Simplify further if needed, but this is the simplified form for the compound fraction.