factor:
a^2/36 - b^2/49
show all your steps please !
To factor the expression (a^2/36 - b^2/49), we need to find the common factors in the numerators and denominators of each term. Then, we can factor out the common factors to simplify the expression.
Step 1: Write the expression with a common denominator.
To do this, we need to find the least common multiple (LCM) of 36 and 49, which is 1764. Multiplying the numerator and denominator of the first term by 49, and the numerator and denominator of the second term by 36, we get:
(49*a^2)/1764 - (36*b^2)/1764
Step 2: Combine the terms.
Now that we have a common denominator, we can combine the two terms into a single fraction:
(49*a^2 - 36*b^2)/1764
Step 3: Look for common factors in the numerator.
The numerator (49*a^2 - 36*b^2) can be factored using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).
Using this formula, we can factor our numerator as:
((7*a)^2 - (6*b)^2)/1764
Step 4: Apply the difference of squares formula.
Using the difference of squares formula, we can factor further:
((7*a + 6*b)(7*a - 6*b))/1764
Step 5: Simplify and check for common factors.
Finally, we simplify the expression. Notice that both terms in the denominator and both terms in the numerator have a common factor of 7:
((a + (6/7)*b)(a - (6/7)*b))/(one or (7*36))
Therefore, the fully factored expression is:
(a + (6/7)*b)(a - (6/7)*b)/(one or 252).