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Chapter 7: Problem 3

Find the least common denominator of the rational expressions. $$\frac{8}{15 x^{2}} \text { and } \frac{5}{6 x^{5}}$$

Short Answer

Expert verified
The least common denominator of \(\frac{8}{15x^{2}}\) and \(\frac{5}{6x^{5}}\) is \(30x^{5}\).

Step by step solution

01

- Factorization

Start by factorizing the denominators of the fractions. Additionally, it can be observed that 15 and 6 (the constants in the denominators) are both divisible by 3, and \(x^{5}\) contains \(x^{2}\): \(15x^{2} = 3 \cdot 5 \cdot x^{2}\) \(6x^{5} = 3 \cdot 2 \cdot x^{2} \cdot x^{3}\)
02

- Identify Common and Individual Factors

Identify the common factors and the factors that are individual to each denominator: Common factors: 3, \(x^{2}\) Individual Factors: 5, 2, \(x^{3}\)
03

- Compute the Least Common Denominator

The LCD is the product of all the common factors and the individual factors of the denominators. Therefore: \(LCD = 3 \cdot 5 \cdot 2 \cdot x^{2} \cdot x^{3} = 30x^{5}\)

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-4}-\frac{4}{4-y}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-3}{y^{2}-25}+\frac{y-3}{25-y^{2}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x-y}+\frac{y}{y-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x}+4$$
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