Problem 103
Question
Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\)
Step-by-Step Solution
Verified Answer
The least common denominator is \( (x+10)(x-10)^{2} \)
1Step 1: Factorizing \(x^{2}-100\)
This is a difference of squares. It can be factorized into \((x+10)(x-10)\)
2Step 2: Factorizing \(x^{2}-20 x+100\)
This is a perfect square trinomial, it can be factorized into \((x-10)^2\)
3Step 3: Getting the least common denominator
To get the least common denominator, get the product of the highest power of each factor from all the denominators. Here, the highest power of \((x+10)\) is 1 which is from \(x^{2}-100\), and the highest power of \((x-10)\) is 2 which is from \(x^{2}-20x+100\). So therefore, the least common denominator is \((x+10)(x-10)^{2}\)
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