Hello, I'd like to know how in developing, factoring or simplifying, we can move from:
A + ( (1 / ( (1 / ICD) + IED) )
to:
ACD *( ( - ED + I ) / (1 - ECD^2 ) )
ACDE are strictly positive real constants.
I is an imaginary number.
I don't get it either. Parentheses are mismatched, and I get this.
A + CD/(1-ECD^2) I
Check for typos. The upper-case is annoying, too.
I made a picture.
goo.gl/zZdzSs
To move from the expression A + ((1 / ((1 / ICD) + IED)) to ACD *((-ED + I) / (1 - ECD^2)), we need to simplify and perform some algebraic manipulations. Here's the step-by-step process:
1. Start with the given expression: A + ((1 / ((1 / ICD) + IED))
2. Simplify the expression inside the innermost parentheses ((1 / ICD) + IED). Since IED is a positive real constant, we can rewrite it as I * ECD. The expression becomes: A + ((1 / ((1 / ICD) + I * ECD))
3. To simplify further, take the reciprocal of the entire expression inside the outermost parentheses (1 / ((1 / ICD) + I * ECD)). This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of ((1 / ICD) + I * ECD) is ((1 / ICD) - I * ECD).
Multiply numerator and denominator by the conjugate: A * ((1 / ICD) - I * ECD) / (((1 / ICD) + I * ECD) * ((1 / ICD) - I * ECD))
Simplify the denominator: A * ((1 / ICD) - I * ECD) / (1 / ICD)^2 - (I * ECD)^2
Simplify further: A * ((1 / ICD) - I * ECD) / (1 / (ICD^2)) - (-1 * ECD^2)
Combine fractions: A * ((1 / ICD) - I * ECD) / (1 / (ICD^2)) + ECD^2
Simplify: A * ((1 / ICD) - I * ECD) * (ICD^2) / 1 + (ECD^2 * ICD^2)
This gives: A * ((ICD - I * ECD) / (1 + ECD^2 * ICD^2))
4. Now, we can multiply the numerator and denominator of the expression by ACD, since ACD is a positive real constant:
Multiply numerator and denominator by ACD: (A * ACD * (ICD - I * ECD)) / (ACD * (1 + ECD^2 * ICD^2))
Simplify further: ACD * A * (ICD - I * ECD) / (ACD + ACD * ECD^2 * ICD^2)
5. Rearrange the terms in the numerator: ACD * A * (ICD - I * ECD) = A * ACD * (ICD - I * ECD)
The final expression becomes: A * ACD * (ICD - I * ECD) / (ACD + ACD * ECD^2 * ICD^2)
Now, the expression has been simplified to ACD *((-ED + I) / (1 - ECD^2)).