Problem 145 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Establish the equations

We have three equations from the exercise: \(b^{A}=M N, b^{C}=M,\) and \(b^{D}=N\). The aim is to identify the relationship between \(A\), \(C\), and \(D\)

Step 2 - Equation manipulation using exponent rules

Using the rule of exponents, \(b^{A}=M N\) can be rewritten as \(b^{A}=b^{C} b^{D}\). In terms of exponents with the same bases, we know that when multiplying terms with the same base, the exponents are added. So, \(b^{A}=b^{C+D}\)

Step 3 - Conclusion of relationship

The final step is comparing the two expressions from step 2. Given that the bases \(b\) are identical on both sides of the equation, this implies that their exponents are equivalent, i.e. \(A = C + D\)