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Chapter 0: Problem 93

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\).

Short Answer

Expert verified
The least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20x+100\) is \((x-10)^{2}(x+10)\).

Step by step solution

01

Factorize the Expressions

Begin by factorizing the denominators. The difference of two squares, \(x^{2}-100\), can be factored into \((x-10)(x+10)\). The perfect square trinomial, \(x^{2}-20x+100\), can be factored into \((x-10)^{2}\).
02

Identify Common Factors

After factorizing the denominators, identify the highest power of each factor in both expressions. In this example, the factor \(x-10\) appears as \((x-10)\) in the first expression and \((x-10)^{2}\) in the second.
03

Form the LCD

The LCD is the product of the highest power of all identified factors. So, the LCD for this problem would be \((x-10)^{2}(x+10)\), since the highest power of \(x-10\) found was 2, and \(x+10\) was also a factor in the expressions.

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Most popular questions from this chapter

Perform the indicated operations. $$ [(3 x+y)+1]^{2} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ x^{3}-64-(x+4)\left(x^{2}+4 x-16\right) $$

Simplify using properties of exponents. $$\frac{20 x^{\frac{1}{2}}}{5 x^{4}}$$

Describe the kinds of numbers that have rational fifth roots.

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.
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