To simplify an exponential expression using the Quotient Rule of Integer Exponents, follow these steps:
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Identify the Expression: Make sure you have a fraction where both the numerator and the denominator are powers with the same base. It will look like this: \(\frac{a^m}{a^n}\), 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 Integer Exponents, when you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The rule is expressed as: \[ \frac{a^m}{a^n} = a^{m-n} \]
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Simplify the Expression: After subtracting the exponents, write the new expression. If \(m - n\) results in a negative exponent, you can express it as: \[ a^{m-n} = \frac{1}{a^{n-m}} \] if needed.
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Final Check: Ensure that the simplified expression adheres to any restrictions (e.g., the base \(a\) should not be zero) and that the answer is in its simplest form.
Example:
Simplify the expression \(\frac{x^5}{x^2}\):
- Identify: The expression is \(\frac{x^5}{x^2}\).
- Apply the Quotient Rule: Subtract the exponents: \(5 - 2 = 3\).
- Simplify: Therefore, \(\frac{x^5}{x^2} = x^{5-2} = x^3\).
In summary, when simplifying using the Quotient Rule, always remember the key step of subtracting the exponents of the same base in the fraction.
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