A ratio of two algebraic expressions a/b for example, where b ≠ 0, is called a
rational
algebraic expressions.
In the ratio a/b ,a is called the numerator, b
is the denominator and is
not equal to zero.
Thus, an algebraic fraction is said to be
defined when its denominator is not equal to zero.
Examples of Algebraic
Fractions:
2x , -7x , -5x , -4x
3y 5y 6y 8y
NOTE: 3y, 5y, 6y, and 8y
are not equal to zero. Here, we may refer to any rational algebraic expression
simply as an algebraic fraction or just fraction.
Simplifying a fraction means reducing
it to its lowest term; and it is said to be on its lowest term if both the
numerator and the denominator have no common factor except 1 and -1.
To simplify fraction, divide the numerator
and the denominator by the HCF
( highest common factor ),
or factor both the numerator and the denominator into prime factors and divide
the common factors. The HCF is the expression of the highest degree that can be
the divisor of both the numerator and the denominator.
Illustrative Examples:
1. Simplify: 4x2y7z
12x5y3
Solution: Find first the
HCF, and that is 4x2y3. Then divide the numerator and the denominator by the
HCF.
4x2y7z
= 4x2y3 ( y4z
) = y4z
12x5y3 4x2y3 (
3x3 ) 3x3
2. Simplify: 4x4y5
6x6y4
Solution:
4x4y5
= 2x4y4 ( 2y ) =
2y
6x6y4 2x4y4 (
3x2 ) 3x2
3. Simplify:
a2 - b2
(a+b)(3a-b)
Solution:
|
|
a2 – b2 (a+b) (a-b) a – b
(a+b)(3a-b) (a+b) (3a-b) 3a-b
NOTE: a2 – b2
is a difference of two squares, and a –
b cannot be factored anymore.
3a
- b
Fractions that show the same values are
called equivalent fractions. These
are fractions which can be interchanged with each other by factoring or by
changing to lowest terms or to higher multiples. Examples are the ff:
1/2 and 3/6, x^2/y^3 and x^3/xy^3, a/2a and 5a/10a
The operations on algebraic fractions
follow the same rule in arithmetic.
Just like in the arithmetic, only similar
terms or like fractions ( meaning, having the same denominators ) can be added
or subtracted, and the dissimilar or unlike fractions ( meaning, having
different denominators ) can only be indicated.
Examples:
This one is more complex than the one
having similar fractions. The following are the steps in adding or subtracting
dissimilar fractions:
- Find the Least Common Denominator ( LCD ). Factor
if necessary.
- Change the fractions to similar fraction using the
LCD.
- Add or subtract the numerator and write the result
over the LCD.
- Always express the answer in its simplest form.
NOTE: The least common
denominator ( lcd ) is the smallest number that is divisible by the
denominators of the fractions involved. In order to find the lcd, just look for
the least common multiples ( lcm ) of the fractions denominator.
To find the product of two or more fractions, just multiply their
numerators and their denominators and
express the result in its simplest form. Another way is by using the
process of multiplicative cancellation.
That is by factoring first the given ( numerator and denominator ) if possible
and then cancell the factors.
Illustrative
Examples:
·
1. Multiply:
x – 4
4x + 8
2x + 8 x2 – 16
To divide fractions, multiply the dividend by the reciprocal of the
divisor. The reciprocal of a
fraction is the inverse of the faction. In symbol
A complex fraction is a fraction that contains one or more fraction
in either its numerator or denominator. It maybe simplified by reducing its
numerator and denominator to single fraction and then divide and reduce the
result to lowest term.
Another way is by getting the LCD of all fractions contained in the
complex fraction and multiply it to the numerator and the denominator of the
said fraction, then simplify.
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