Complex Fractions
_1_ - _2_
x^2 3
____________
_1_ - _5_
x 6
3 is supposed to be the denominator of 2
6 is supposed to be the denominator of 5
So I see it as
(1/x^2 - 2/3) / (1/x - 5/6)
= (3-2x^2)/(3x^2) / (6x)/(6-5x)
= (3 - 2x^2)/(3x^2) (6x)/(6-5x)
= 2(3-2x^2)/(x(6-5x))
or
= 2(2x^2 - 3)/(x(5x - 6))
To simplify the complex fraction, we need to find a common denominator for all the fractions involved. In this case, we have denominators of x^2, 3, x, and 6. To find the common denominator, we need to find the least common multiple (LCM) of these denominators.
Step 1: Find the LCM of x^2, 3, x, and 6.
The prime factorization of each number is:
x^2 = x * x
3 = 3
x = x
6 = 2 * 3
The LCM is the product of the highest powers of all the prime factors. So the LCM is:
x^2 * 3 * x * 2 = 6x^3.
Step 2: Rewrite each fraction with the common denominator 6x^3.
For the first fraction:
_1_ _2_
x^2 3
Multiply the numerator and denominator of the first fraction by (2x)(2x^2):
_1_ * (2x)(2x^2) _2_ * (2x)(2x^2)
x^2 * (2x)(2x^2) 3 * (2x)(2x^2)
Simplifying the numerators:
_1_ * 4x^3 _2_ * 4x^3
2 2
The first fraction becomes:
4x^3 / 2.
For the second fraction:
_1_ _5_
x 6
Multiply the numerator and denominator of the second fraction by (x^2)(2):
_1_ * (x^2)(2) _5_ * (x^2)(2)
x * (x^2)(2) 6 * (x^2)(2)
Simplifying the numerators:
_1_ * 2x^3 _5_ * 2x^3
1 1
The second fraction becomes:
2x^3 / 1.
So the complex fraction can be rewritten as:
_4x^3/2_ - _2x^3/1_
____________
1 6
Step 3: Perform the subtraction of the fractions.
To subtract the fractions, we need to find a common denominator for the numerators. Since the denominators are already the same, we can subtract the numerators directly.
(_4x^3/2_) - (_2x^3/1_) = (4x^3 - 2x^3) / 2
Simplifying the numerator:
4x^3 - 2x^3 = 2x^3
So the simplified numerator is 2x^3.
The complex fraction now becomes:
_2x^3_
________
2
6
Step 4: Simplify the result.
To simplify the fraction further, we can divide both the numerator and denominator by their greatest common factor (GCF), which in this case is 2.
Dividing the numerator and denominator by 2:
2x^3 / 2 = (2/2) * (x^3/1) = 1 * x^3 = x^3
So the final simplified form of the complex fraction is just x^3.