If `(x^(a)x^(b))/((x^(c))^(d))=x^(2)` for all `x ne 0`, which of the following must be true?

Options

A. `a+b-cd=2`
B. `(a+b)/(cd)=2`
C. `ab-cd=2`
D. `ab-c^(d)=2`

Answer

To solve the equation \(\frac{x^a \cdot x^b}{(x^c)^d} = x^2\) for all \(x \neq 0\), we can follow these steps:

### Step 1: Simplify the left-hand side
We start by simplifying the left-hand side of the equation. We can use the properties of exponents to combine the terms in the numerator and the denominator.

\[ \frac{x^a \cdot x^b}{(x^c)^d} = \frac{x^{a+b}}{x^{cd}} \]

### Step 2: Apply the quotient rule of exponents
Next, we apply the quotient rule of exponents, which states that \(\frac{x^m}{x^n} = x^{m-n}\).

\[ \frac{x^{a+b}}{x^{cd}} = x^{(a+b) - cd} \]

### Step 3: Set the exponents equal
Now, we set the exponent from the left-hand side equal to the exponent on the right-hand side of the equation, which is 2.

\[ (a + b) - cd = 2 \]

### Step 4: Rearrange the equation
We can rearrange the equation to express a relationship between \(a\), \(b\), \(c\), and \(d\).

\[ a + b - cd = 2 \implies a + b = cd + 2 \]

### Conclusion
Thus, the relationship that must be true for the equation to hold for all \(x \neq 0\) is:

\[ a + b - cd = 2 \]