Simplify \( \frac{u^{-9}}{u^{3}} \) and express your answer using positive exponents.
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Understand the Problem
The question is asking us to simplify the expression ( \frac{u^{-9}}{u^{3}} ), and present the answer using only positive exponents.
Answer
The answer is \( \frac{1}{u^{12}} \).
Answer for screen readers
The simplified expression using only positive exponents is ( \frac{1}{u^{12}} ).
Steps to Solve
- Apply the Quotient Rule for Exponents
Using the quotient rule, when dividing like bases, we subtract the exponent in the denominator from the exponent in the numerator.
So, we have: $$ \frac{u^{-9}}{u^{3}} = u^{-9 - 3} $$
- Simplify the Exponent
Now, simplify the exponent: $$ -9 - 3 = -12 $$
Thus, we get: $$ \frac{u^{-9}}{u^{3}} = u^{-12} $$
- Convert to Positive Exponents
To express with positive exponents, we can rewrite $ u^{-12} $ as: $$ \frac{1}{u^{12}} $$
The simplified expression using only positive exponents is ( \frac{1}{u^{12}} ).
More Information
When simplifying expressions with exponents, it's important to remember the rules for multiplying and dividing like bases, particularly the quotient rule. Converting negative exponents to positive form often involves rewriting the expression as a fraction.
Tips
- Forgetting to subtract the exponents correctly when applying the quotient rule.
- Not converting negative exponents to positive, which is often required in simplifications.
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