Asked by Ous
Find the least common denominator of the rational expressions.
StartFraction 9 Over 10 x squared EndFraction
and StartFraction 7 Over 25 x Superscript 6 EndFraction
StartFraction 9 Over 10 x squared EndFraction
and StartFraction 7 Over 25 x Superscript 6 EndFraction
Answers
Answered by
GPT-4o mini
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.
1. **Identify the denominators**:
- The first denominator is \( 10x^2 \).
- The second denominator is \( 25x^6 \).
2. **Factor each denominator**:
- The factors of \( 10 \) are \( 2 \times 5 \).
- The factors of \( 25 \) are \( 5^2 \).
- Therefore, we can write:
- \( 10 = 2^1 \times 5^1 \)
- \( 25 = 5^2 \)
- Hence, we can express the denominators as:
- \( 10x^2 = 2^1 \times 5^1 \times x^2 \)
- \( 25x^6 = 5^2 \times x^6 \)
3. **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, \( \text{LCM}(10, 25) = 2^1 \times 5^2 = 2 \times 25 = 50 \).
4. **Find the LCM of the variable parts**:
- For \( x^2 \) and \( x^6 \), the highest power is \( x^6 \).
5. **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 \(\boxed{50x^6}\).
1. **Identify the denominators**:
- The first denominator is \( 10x^2 \).
- The second denominator is \( 25x^6 \).
2. **Factor each denominator**:
- The factors of \( 10 \) are \( 2 \times 5 \).
- The factors of \( 25 \) are \( 5^2 \).
- Therefore, we can write:
- \( 10 = 2^1 \times 5^1 \)
- \( 25 = 5^2 \)
- Hence, we can express the denominators as:
- \( 10x^2 = 2^1 \times 5^1 \times x^2 \)
- \( 25x^6 = 5^2 \times x^6 \)
3. **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, \( \text{LCM}(10, 25) = 2^1 \times 5^2 = 2 \times 25 = 50 \).
4. **Find the LCM of the variable parts**:
- For \( x^2 \) and \( x^6 \), the highest power is \( x^6 \).
5. **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 \(\boxed{50x^6}\).