Question
Determine the following: \( a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} \)
Ask by Harris Martinez. in South Africa
Nov 11,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( a \).
Solution
To simplify the expression \( a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} \), we can use the properties of exponents. Specifically, we can combine the terms with the same base by adding their exponents.
First, we will group the terms with base \( a \) and the terms with base \( b \):
1. For the base \( a \):
\[
a^{4} \cdot a^{2} \cdot a^{-5} = a^{4 + 2 - 5} = a^{1} = a
\]
2. For the base \( b \):
\[
b^{-9} \cdot b^{7} \cdot b^{2} = b^{-9 + 7 + 2} = b^{0}
\]
Since \( b^{0} = 1 \), we can now combine the results:
\[
a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} = a \cdot 1 = a
\]
Thus, the simplified expression is:
\[
\boxed{a}
\]
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Simplify this solution Beyond the Answer
Let's simplify the expression \( a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} \) step by step. First, we can combine the terms with the same base (for \(a\) and \(b\)) separately using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). For \(a\): \[ a^{4} \cdot a^{2} \cdot a^{-5} = a^{4 + 2 - 5} = a^{1} = a \] Now for \(b\): \[ b^{-9} \cdot b^{7} \cdot b^{2} = b^{-9 + 7 + 2} = b^{0} \] Since \( b^0 = 1 \), we can now combine our results: \[ a^{4} b^{-9} \cdot a^{2} b^{7} \cdot a^{-5} b^{2} = a^{1} \cdot 1 = a \] Thus, the simplified expression is: \[ \boxed{a} \]
