To simplify an exponential expression using the quotient rule of integer exponents, you should follow these steps:
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Identify the Expression: Ensure that you have a fraction involving exponential terms in the form \(\frac{a^m}{a^n}\), where \(a\) is the base, and \(m\) and \(n\) are integers (the exponents).
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Apply the Quotient Rule: According to the quotient rule for exponents, when you divide two exponential expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator: \[ \frac{a^m}{a^n} = a^{m-n} \]
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Perform the Subtraction: Execute the subtraction \(m-n\) to find the new exponent.
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Rewrite the Expression: Rewrite the simplified expression with the new exponent: \[ a^{m-n} \]
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Simplify Further if Necessary: If \(m-n\) can be simplified further (e.g., if it’s a negative exponent), apply the rules for negative exponents if needed: \[ a^{-k} = \frac{1}{a^k} \]
Example
For example, simplify \(\frac{x^5}{x^2}\):
- Identify the expression: \(\frac{x^5}{x^2}\)
- Apply the quotient rule: We apply \(x^{5-2}\).
- Perform the subtraction: \(5 - 2 = 3\).
- Rewrite the expression: This gives us \(x^3\).
- Simplify further if necessary: In this case, no further simplification is needed.
The final simplified result is \(x^3\).
By following these steps systematically, you can simplify any exponential expression using the quotient rule of integer exponents.