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Chapter 7: Problem 1

Find the least common denominator of the rational expressions. $$\frac{7}{15 x^{2}} \text { and } \frac{13}{24 x}$$

Short Answer

Expert verified
The least common denominator of the rational expressions \( \frac{7}{15 x^{2}} \) and \( \frac{13}{24 x} \) is \( 120x^2 \).

Step by step solution

01

Prime Factorization of the Denominators

Find the prime factorization of each denominator. - For 15, the prime factors are \( 3 \times 5 \)- For \(x^{2}\), the prime factors are \( x \times x \)- For 24, the prime factors are \( 2 \times 2 \times 2 \times 3 \)- For \( x \), the prime factor is \( x \)
02

Identify the Highest Powers of the Prime Factors

Identify the highest power of each prime factor from both denominators. - Highest power of 2: \(2^3\)- Highest power of 3: \(3^1\)- Highest power of 5: \(5^1\)- Highest power of \(x\): \(x^2\)
03

Calculate the Least Common Denominator

Calculate the LCD by multiplying the highest powers of all the prime factors. - LCD = \(2^3 \times 3^1 \times 5^1 \times x^2 = 120x^2\).

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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. $$\frac{y}{y-1}-\frac{1}{1-y}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

perform the indicated operations. Simplify the result, if possible. $$\left(\frac{3 x^{2}-4 x+4}{3 x^{2}+7 x+2}-\frac{10 x+9}{3 x^{2}+7 x+2}\right) \div \frac{x-5}{x^{2}-4}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$
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