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Simplify. \( \frac{\frac{c^{2}-d^{2}}{c d}}{\frac{c-d}{d}} \) The simplified expression is \( \square \).

Ask by Goodwin Carrillo. in the United States
Jan 22,2025

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Tutor-Verified Answer

Answer

The simplified expression is \( \frac{c + d}{c} \).

Solution

To simplify the expression: \[ \frac{\frac{c^{2}-d^{2}}{c d}}{\frac{c-d}{d}} \] **Step 1:** Rewrite the complex fraction as a multiplication: \[ \frac{c^{2} - d^{2}}{c d} \times \frac{d}{c - d} \] **Step 2:** Simplify the expression by canceling out common terms. Notice that \(c^2 - d^2\) is a difference of squares, which can be factored: \[ c^2 - d^2 = (c + d)(c - d) \] Substitute this back into the expression: \[ \frac{(c + d)(c - d)}{c d} \times \frac{d}{c - d} \] **Step 3:** Cancel out the \((c - d)\) terms and the \(d\) terms: \[ \frac{c + d}{c} \] **Final Answer:** \[ \boxed{\dfrac{c + d}{\,c\,}} \]

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To simplify the expression \( \frac{\frac{c^{2}-d^{2}}{c d}}{\frac{c-d}{d}} \), start by recognizing that \( c^2 - d^2 \) can be factored using the difference of squares: \[ c^2 - d^2 = (c - d)(c + d) \] Now, substituting this into the expression gives: \[ \frac{\frac{(c - d)(c + d)}{c d}}{\frac{c - d}{d}} \] Next, we can rewrite the fraction in the denominator: \[ \frac{(c - d)(c + d)}{c d} \div \frac{c - d}{d} = \frac{(c - d)(c + d)}{c d} \times \frac{d}{c - d} \] Notice that \( c - d \) in the numerator and denominator cancels out (assuming \( c \neq d \)): \[ = \frac{(c + d)}{c} \] The final simplified expression is: \[ = \frac{c + d}{c} \] Thus, the result is: \[ \frac{c + d}{c} \] The simplified expression is \( \frac{c + d}{c} \).

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