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3. \( c^{-5} d^{7} \)

Ask by Rodriquez Higgins. in the United States
Jan 27,2025

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Answer

The simplified form of \( c^{-5} d^{7} \) is \( \frac{d^{7}}{c^{5}} \).

Solution

It looks like you're working with the expression: \[ c^{-5} d^{7} \] To simplify this expression by eliminating the negative exponent, you can rewrite it with positive exponents. Here's how you can do it: ### Step-by-Step Solution: 1. **Understand Negative Exponents:** A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Specifically: \[ c^{-5} = \frac{1}{c^{5}} \] 2. **Rewrite the Expression:** Replace \( c^{-5} \) with its reciprocal form: \[ c^{-5} d^{7} = \frac{1}{c^{5}} \times d^{7} \] 3. **Combine the Terms:** You can express the entire expression as a single fraction: \[ \frac{d^{7}}{c^{5}} \] ### Final Simplified Form: \[ \frac{d^{7}}{c^{5}} \] This represents the original expression with all exponents positive.

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The Deep Dive

Did you know that negative exponents actually indicate a reciprocal? For your expression \( c^{-5} d^{7} \), you can rewrite \( c^{-5} \) as \( \frac{1}{c^{5}} \), transforming it to \( \frac{d^{7}}{c^{5}} \). This makes calculations clearer and more straightforward, especially in algebra! In real-world applications, manipulating expressions with exponents is crucial, particularly in fields like physics or engineering. For instance, when dealing with formulas for exponential growth or decay, simplifying complex expressions gives clarity. It's like decluttering your workspace—when it's tidy, you can focus on designing those impressive structures or analyzing exciting data without the extra chaos!

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