Question 425608
To find the LCD we need to know what the factors of the denominators are. So we start by factoring the denominators. (From this point on I am going to ignore the numerators since they have no impact of what the LCD is or on how we find the LCD.)<br>
The first denominator is a difference of squares so we can use the {{{a^2-b^2 = (a+b)(a-b)}}} pattern to factor it:
{{{h^2-k^2 = (h+k)(h-k)}}}<br>
The second denominator is a perfect trinomial square so we can use the {{{a^2-2ab+b^2 = (a-b)(a-b)}}} pattern tp factor it:
{{{h^2-2hk+k^2 = (h-k)(h-k)}}}<br>
So our two denoinators, in factored form are:
(h+k)(h-k) and (h-k)(h-k)<br>
The LCD will be the smallest product that includes all the factors of both denominators. In this case the LCD will be:
(h+k)(h-k)(h-k)
Looking at the LCD you can see the first denominator in the first two factors and the second denominator in the last two factors.<br>
(h+k)(h-k)(h-k)
is the LCD and this may be an acceptable form. If not, then multiply this out. Multiplying the first two factors is easy because they cam from the first denominator:
{{{(h^2-k^2)(h-k)}}}
Now we just use FOIL:
{{{h^3-h^2k-hk^2-k^3}}}<br>
So the LCD is either
(h+k)(h-k)(h-k)
or
{{{h^3-h^2k-hk^2+k^3}}}
depending on the desired form.