To simplify the expression \( \frac{t^{400} \cdot t^{100} \cdot t^{4}}{t^{300} \cdot t^{500}} \) using the Quotient Rule of Integer Exponents, we can follow these steps:
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Combine the exponents in the numerator and the denominator:
- For the numerator: \[ t^{400} \cdot t^{100} \cdot t^{4} = t^{400 + 100 + 4} = t^{504} \]
- For the denominator: \[ t^{300} \cdot t^{500} = t^{300 + 500} = t^{800} \]
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Now the expression looks like: \[ \frac{t^{504}}{t^{800}} \]
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Applying the Quotient Rule, which states that \( \frac{a^m}{a^n} = a^{m-n} \), we get: \[ t^{504 - 800} = t^{-296} \]
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The final simplified expression is: \[ t^{-296} \]
If you want to express it positively, you can also write it as: \[ \frac{1}{t^{296}} \]
So, the simplified expression is \( t^{-296} \) or \( \frac{1}{t^{296}} \).