simlify the following to the lowest terms: (a) [x^2-y^2]/[x^2+y^2] (b) [3a^2+6ab-a-2b]/[9a^2-6a+1] (c) [x^4-y^4]/[x^3-y^3] (d) [36x^3y^4z]/[64x^5y^2z^3]
To simplify rational expressions to the lowest terms, we need to find the greatest common factor (GCF) of the numerator and the denominator and then divide both by it. I will explain how to simplify each of the given expressions step by step.
(a) [x^2-y^2]/[x^2+y^2]:
To simplify this expression, we notice that the numerator is a difference of squares, which can be factored as (x+y)(x-y). The denominator does not factor further. Therefore, our expression becomes:
[x^2-y^2]/[x^2+y^2] = [(x+y)(x-y)]/[x^2+y^2]
This expression cannot be simplified further since the numerator and denominator do not have any common factors.
(b) [3a^2+6ab-a-2b]/[9a^2-6a+1]:
To simplify this expression, we can first try factoring the numerator and denominator if possible. However, in this case, the numerator and denominator do not have any common factors that can be factored out. Therefore, this expression is already in its simplest form.
(c) [x^4-y^4]/[x^3-y^3]:
This expression is a difference of squares in the numerator and the denominator. We can factor it using the difference of squares formula:
[x^4-y^4] = (x^2+y^2)(x^2-y^2)
[x^3-y^3] = (x-y)(x^2+xy+y^2)
Now, our expression becomes:
[x^4-y^4]/[x^3-y^3] = [(x^2+y^2)(x^2-y^2)]/[(x-y)(x^2+xy+y^2)]
We can further simplify by canceling the common factor of (x^2-y^2):
[x^4-y^4]/[x^3-y^3] = (x^2+y^2)/(x^2+xy+y^2)
Therefore, this is the simplified form of the given expression.
(d) [36x^3y^4z]/[64x^5y^2z^3]:
To simplify this expression, we need to consider the common factors in the numerator and denominator. Here, the GCF of the numerator is 4xy^2z and the GCF of the denominator is 8x^3y^2z^2. Dividing both the numerator and denominator by their respective GCF will result in the simplified expression:
[36x^3y^4z]/[64x^5y^2z^3] = (4xy^2z)/(8x^3y^2z^2)
Now, we can simplify the expression by canceling common factors:
(4xy^2z)/(8x^3y^2z^2) = 1/(2x^2z)
Therefore, this is the simplified form of the given expression.