To find the least common denominator (LCD) of the rational expressions \( \frac{9}{10x^2} \) and \( \frac{7}{25x^6} \), we need to determine the least common multiple (LCM) of the denominators.
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Identify the denominators:
- The first denominator is \( 10x^2 \).
- The second denominator is \( 25x^6 \).
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Factor each denominator:
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The factors of \( 10 \) are \( 2 \times 5 \).
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The factors of \( 25 \) are \( 5^2 \).
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Therefore, we can write:
- \( 10 = 2^1 \times 5^1 \)
- \( 25 = 5^2 \)
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Hence, we can express the denominators as:
- \( 10x^2 = 2^1 \times 5^1 \times x^2 \)
- \( 25x^6 = 5^2 \times x^6 \)
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Find the LCM of the numerical parts (coefficients):
- The LCM of \( 10 \) and \( 25 \):
- For the prime factor \( 2 \): The highest power is \( 2^1 \).
- For the prime factor \( 5 \): The highest power is \( 5^2 \).
- Thus, \( \text{LCM}(10, 25) = 2^1 \times 5^2 = 2 \times 25 = 50 \).
- The LCM of \( 10 \) and \( 25 \):
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Find the LCM of the variable parts:
- For \( x^2 \) and \( x^6 \), the highest power is \( x^6 \).
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Combine the results to get the LCD:
- The least common denominator is obtained by multiplying the LCM of the numerical parts and the LCM of the variable parts: \[ \text{LCD} = 50x^6 \]
Thus, the least common denominator of the rational expressions \( \frac{9}{10x^2} \) and \( \frac{7}{25x^6} \) is \(\boxed{50x^6}\).