Final answer:
The least common denominator of the expressions 9/4x²-12x and 7x/x²-9 is 4x(x-3)(x+3). This is found by factoring each denominator and including each factor in the highest power.
Step-by-step explanation:
The least common denominator (LCD) of the two rational expressions 9/4x²-12x and 7x/x²-9 must accommodate all factors present in both denominators.
First, factor each denominator where possible. The first denominator, 4x²-12x, can be factored by taking out the common factor of 4x, giving us 4x(x-3). The second denominator, x²-9, is a difference of squares and can be factored into (x+3)(x-3).
Now we can identify the LCD by looking at each unique factor. The factors we have are 4x, (x-3), and (x+3). The LCD must include each factor in the highest power it appears. Since 4x and (x-3) come from the first denominator and (x+3) comes from the second, combined they form the LCD, so we have:
LCD = 4x(x-3)(x+3)
When finding an LCD, we eliminate terms by including each factor once and in its highest occurring power only which simplifies the algebra. Always remember to check that your final answer is reasonable and includes all necessary factors.