Question
\( 1.3 \quad \frac { ( x + 1 ) ^ { 3 } ( x - 1 ) ^ { - 3 } } { ( x + 1 ) ^ { 2 } ( x - 1 ) ^ { - 2 } } \)
Ask by Chadwick Lawson. in South Africa
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{x+1}{x-1} \).
Solution
We start with the expression
\[
\frac{(x+1)^3 (x-1)^{-3}}{(x+1)^2 (x-1)^{-2}}.
\]
**Step 1.** Simplify the factors with the base \(x+1\) by subtracting the exponent in the denominator from the numerator:
\[
(x+1)^{3-2} = (x+1)^1 = x+1.
\]
**Step 2.** Simplify the factors with the base \(x-1\) by subtracting the exponent in the denominator from the numerator:
\[
(x-1)^{-3-(-2)} = (x-1)^{-3+2} = (x-1)^{-1}.
\]
Recall that a negative exponent means taking the reciprocal:
\[
(x-1)^{-1} = \frac{1}{x-1}.
\]
**Step 3.** Combine the simplified parts:
\[
(x+1) \cdot \frac{1}{x-1} = \frac{x+1}{x-1}.
\]
Thus, the simplified expression is
\[
\frac{x+1}{x-1}.
\]
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Simplify this solution Extra Insights
To simplify the expression, let’s break it down. We have: \[ \frac{ (x + 1)^{3} (x - 1)^{-3} }{ (x + 1)^{2} (x - 1)^{-2} } \] We can simplify the fractions by applying the exponent rules. When dividing like bases, we subtract the exponents: 1. For \((x + 1)\), we get \((x + 1)^{3 - 2} = (x + 1)^{1} = x + 1\). 2. For \((x - 1)\), we get \((x - 1)^{-3 - (-2)} = (x - 1)^{-1} = \frac{1}{(x - 1)}\). Now combining these results, the expression simplifies to: \[ \frac{x + 1}{x - 1} \] Thus, the final simplified expression is: \[ 1.3 \quad \frac{x + 1}{x - 1} \]
