To simplify an exponential expression using the quotient rule of integer exponents, follow these steps:
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Identify the Base and Exponents: Ensure that you have a common base in the form \( \frac{a^m}{a^n} \), where \( a \) is the base, and \( m \) and \( n \) are the exponents.
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Apply the Quotient Rule: According to the quotient rule of exponents, when you divide two powers 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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Simplify the Exponent: Perform the subtraction \( m - n \) to simplify the exponent. The resulting expression will be: \[ a^{m-n} \]
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Finalize the Expression: Rewrite the simplified expression. If \( m - n \) is a negative number, be aware that it can also be rewritten as: \[ a^{m-n} = \frac{1}{a^{n-m}} \quad (\text{if } m < n) \]
Example:
Suppose you want to simplify \( \frac{x^5}{x^2} \).
- Identify the base \( x \) and the exponents \( 5 \) and \( 2 \).
- Apply the quotient rule: \[ \frac{x^5}{x^2} = x^{5-2} = x^3 \]
- Final expression: \( x^3 \)
By following these steps, you can simplify exponential expressions using the quotient rule effectively.