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Chapter 0: Problem 142

If \(b^{A}=M N, b^{C}=M,\) and \(b^{D}=N,\) what is the relationship among \(A, C,\) and \(D ?\)

Short Answer

Expert verified
The relationship among A, C, and D is \(A = C + D\).

Step by step solution

01

Combine the given equations

Using the given equations, \(b^{C}=M\) and \(b^{D}=N\), substitute \(M\) and \(N\) into the first given equation. Hence, the equation becomes \(b^{A}=b^{C}b^{D}\).
02

Apply the Exponential Laws

According to the law of exponents for multiplication, when the bases of exponential expressions being multiplied together are the same, the exponents should be added. Hence, we can express the equation as \(b^{A}=b^{C+D}\).
03

Equate the Exponents

As the bases are the same, the exponents should also be equal. Therefore, \(A\) is equal to the sum of \(C\) and \(D\), i.e. \(A = C + D\).

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