Final answer:
The equivalent expressions with the LCD for (8)/(14x^4) and (y)/(28x^2) are (16x^2)/(28x^4) and (14xy^2)/(28x^4).
Step-by-step explanation:
To find equivalent expressions that have the least common denominator (LCD) for (8)/(14x^4) and (y)/(28x^2), we need to identify the common factors in the denominators and find the lowest power for each factor.
- The prime factorization of 14x^4 is 2 * 7 * x^4.
- The prime factorization of 28x^2 is 2 * 2 * 7 * x^2.
To find the LCD, we take the highest power of each prime factor that appears in either denominator. In this case, the highest power of 2 is 2, the highest power of 7 is 1, and the highest power of x is 4.
Therefore, the LCD is 2 * 7 * x^4.
To create equivalent expressions with this LCD, we need to multiply each fraction by a factor that will result in the LCD in the denominator. Let's go through the steps:
1. For the fraction (8)/(14x^4), we need to multiply the numerator and denominator by the missing factors: 2 * x^2.
This gives us: (8 * 2 * x^2) / (14 * x^4 * 2 * x^2).
Simplifying, we get: (16x^2) / (28x^4).
2. For the fraction (y)/(28x^2), we need to multiply the numerator and denominator by the missing factors: 2 * 7 * x^2.
This gives us: (y * 2 * 7 * x^2) / (28x^2 * 2 * 7 * x^2).
Simplifying, we get: (14xy^2) / (28x^4).
So, the equivalent expressions with the LCD of 2 * 7 * x^4 are:
- (16x^2) / (28x^4)
- (14xy^2) / (28x^4)
These expressions have the same value as the original fractions but have a common denominator.
In summary, the equivalent expressions with the LCD for (8)/(14x^4) and (y)/(28x^2) are (16x^2)/(28x^4) and (14xy^2)/(28x^4).