Question
Which expression is equivalent to the given expression? Assume the denominator does not equal zero. \[ \left(\frac{3 c^{2} d^{4}}{2 c^{3} d^{3}}\right)^{3} \]
Ask by Wheeler Lyons. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{27d^{3}}{8c^{3}} \).
Solution
To simplify the given expression:
\[
\left(\frac{3 c^{2} d^{4}}{2 c^{3} d^{3}}\right)^{3}
\]
**Step 1: Simplify the Fraction Inside the Parentheses**
\[
\frac{3 c^{2} d^{4}}{2 c^{3} d^{3}} = \frac{3}{2} \cdot \frac{c^{2}}{c^{3}} \cdot \frac{d^{4}}{d^{3}}
\]
Simplify each part:
\[
\frac{c^{2}}{c^{3}} = c^{2-3} = c^{-1} = \frac{1}{c}
\]
\[
\frac{d^{4}}{d^{3}} = d^{4-3} = d^{1} = d
\]
So the expression becomes:
\[
\frac{3}{2} \cdot \frac{1}{c} \cdot d = \frac{3d}{2c}
\]
**Step 2: Raise the Simplified Fraction to the Power of 3**
\[
\left(\frac{3d}{2c}\right)^{3} = \frac{3^{3} d^{3}}{2^{3} c^{3}} = \frac{27d^{3}}{8c^{3}}
\]
**Final Answer:**
\[
\boxed{\dfrac{27\,d^{3}}{8\,c^{3}}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To simplify the expression \(\left(\frac{3 c^{2} d^{4}}{2 c^{3} d^{3}}\right)^{3}\), we first simplify the fraction inside the parentheses. The expression in the fraction is: \[ \frac{3 c^{2} d^{4}}{2 c^{3} d^{3}} = \frac{3}{2} \cdot \frac{c^{2}}{c^{3}} \cdot \frac{d^{4}}{d^{3}} = \frac{3}{2} \cdot \frac{1}{c} \cdot d \] So, we can rewrite it as: \[ \frac{3d}{2c} \] Next, we raise this simplified fraction to the power of 3: \[ \left(\frac{3d}{2c}\right)^{3} = \frac{(3d)^{3}}{(2c)^{3}} = \frac{27d^{3}}{8c^{3}} \] Thus, the expression equivalent to the original expression is: \[ \frac{27d^{3}}{8c^{3}} \]
