Simplify the expression rs^0 · r^4 · s^{-1} · t^{-1} · r^0 · s^t · r^{-9}. Express your answer using positive exponents.
Understand the Problem
The question is asking to simplify the expression involving variables with exponents and then express the final answer using only positive exponents. This involves applying the rules of exponents.
Answer
The simplified expression is \( \frac{s^{t-1}}{r^5 t} \).
Answer for screen readers
The simplified expression is ( \frac{s^{t-1}}{r^5 t} ).
Steps to Solve
- Identify and group terms First, recognize that the expression contains multiple instances of the same variable. Let's group the terms for each variable:
- For ( r ): ( r^0, r^4, r^0, r^{-9} )
- For ( s ): ( s^0, s^{-1}, s^t )
- For ( t ): ( t^{-1} )
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Simplify the exponent for ( r ) Now, we add the exponents of ( r ): [ r^{0 + 4 + 0 - 9} = r^{-5} ]
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Simplify the exponent for ( s ) Next, simplify the exponent for ( s ): [ s^{0 - 1 + t} = s^{-1 + t} ]
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Combine and simplify the ( t ) term The ( t ) variable term remains: [ t^{-1} ]
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Write the terms with positive exponents To express all variables with positive exponents, we can rewrite ( r^{-5} ), ( s^{-1 + t} ), and ( t^{-1} ):
[ \frac{s^{t-1}}{r^5 t} ]
The simplified expression is ( \frac{s^{t-1}}{r^5 t} ).
More Information
In mathematical expressions, exponents can be simplified by combining like terms, and negative exponents indicate that the variable should be placed in the denominator. This technique helps in expressing the final answer concisely.
Tips
- Not combining like terms correctly: Be sure to add/subtract the exponents accurately for each variable.
- Forgetting negative exponents: Remember that negative exponents indicate that the term should be moved to the denominator.
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