Question
Simplify. \( \frac{\frac{c^{2}-d^{2}}{\frac{c d}{c-d}}}{d} \) The simplified expression is \( \square \).
Ask by Savage Klein. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{(c - d)^2 (c + d)}{c d^2} \).
Solution
We start with the expression
\[
\frac{\frac{c^{2}-d^{2}}{\frac{c d}{c-d}}}{d}.
\]
**Step 1.** Recognize the difference of squares:
\[
c^2-d^2 = (c-d)(c+d).
\]
Substitute this into the numerator:
\[
\frac{\frac{(c-d)(c+d)}{\frac{c d}{c-d}}}{d}.
\]
**Step 2.** Simplify the inner fraction by multiplying by the reciprocal:
\[
\frac{(c-d)(c+d)}{\frac{c d}{c-d}} = (c-d)(c+d) \times \frac{c-d}{c d} = \frac{(c-d)^2(c+d)}{c d}.
\]
**Step 3.** Now divide by \( d \):
\[
\frac{\frac{(c-d)^2 (c+d)}{c d}}{d} = \frac{(c-d)^2(c+d)}{c d} \times \frac{1}{d} = \frac{(c-d)^2(c+d)}{c\, d^2}.
\]
Thus, the simplified expression is
\[
\frac{(c-d)^2(c+d)}{c\, d^2}.
\]
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Simplify this solution The Deep Dive
To simplify the expression \( \frac{\frac{c^{2}-d^{2}}{\frac{c d}{c-d}}}{d} \), first, notice that \( c^2 - d^2 \) can be factored as \( (c-d)(c+d) \). Therefore, we can rewrite the expression as: \[ \frac{(c-d)(c+d)}{\frac{c d}{c-d}} \div d \] Now simplifying the inner fraction \( \frac{(c-d)(c+d)}{\frac{c d}{c-d}} \): \[ = \frac{(c-d)(c+d) \cdot (c-d)}{c d} \] This gives us: \[ = \frac{(c-d)^2(c+d)}{c d} \] Now, we will divide this by \( d \): \[ \frac{(c-d)^2(c+d)}{c d} \div d = \frac{(c-d)^2(c+d)}{c d^2} \] Thus, the simplified expression is: \[ \frac{(c-d)^2(c+d)}{c d^2} \] So, the expression is \( \frac{(c-d)^2(c+d)}{c d^2}. \)
