Fractions and their reduction is another topic that begins in the 5th grade. Here the base of this action is formed, and then these skills stretch to the higher mathematics. If the student has not learned how to reduce fractions, then he may have problems in algebra. Therefore, it is better to understand several rules once and for all. And remember one ban and never break it.
Fraction and its reduction
What is it, every student knows. Any two digits located between the horizontal bar are immediately perceived as a fraction. However, not everyone understands that it can be any number. If it is an integer, then it can always be divided into one, then the wrong fraction will be obtained. But more on that later.
The beginning is always simple. First you need to figure out how to cut the correct fraction. That is, one whose numerator is smaller than the denominator. To do this, we need to recall the main property of the fraction. It states that when multiplying (as well as dividing) its numerator and denominator simultaneously by an equal number, an equivalent fraction is obtained.
The actions of division that are performed in this property and lead to a reduction. That is, its maximum simplification. The fraction can be reduced until there are common multipliers above the line and below it. When they are no longer there, the reduction is impossible. And they say that this fraction is irreducible.
1. Step-by-step reduction. It uses the method of calculating when both numbers are divided by the minimum common multiplier that the student noticed. If after the first cut you see that this is not the end, then the division continues. While the fraction does not become irreducible.
2. Finding the greatest common divisor for the numerator and denominator. This is the most rational way of how to reduce fractions. It implies the factorization of the numerator and the denominator into prime factors. Among them, then you need to choose all the same. Their product will give the greatest common factor, to which the fraction is reduced.
Both methods are equivalent. The student is encouraged to master them and use the one that they liked best.
What if there are letters and addition and subtraction actions?
With the first part of the question, everything is more or less clear. Letters can be abbreviated as well as numbers. The main thing is that they act as multipliers. But with the second many people have problems.
It is important to remember! You can only reduce numbers that are multipliers. If they are summands - it is impossible.
In order to understand how to reduce fractions, having the form of an algebraic expression, you need to learn the rule. First, to represent the numerator and denominator as a product. Then you can reduce, if the total multipliers. To view multipliers will be useful for such techniques:
- application of shortened multiplication identities.
Moreover, the latter method makes it possible to immediately obtain terms in the form of multipliers. Therefore, it must always be used if a known pattern is visible.
But it's not scary, then there are tasks with degrees and roots. Then it takes courage and learn a couple of new rules.
Expression with degree
Fraction. In the numerator and denominator the product. There are letters and numbers. And they are also raised to a power, which also consists of terms or multipliers. There is something to be frightened of.
In order to understand how to reduce fractions with degrees, you will need to learn two points:
- if in the exponent there is a sum, then it can be decomposed into factors whose powers are the original summands;
- if the difference, then the dividend and the divisor, the first in the power will be decreasing, the second - subtracted.
After performing these actions, common multipliers become visible. In such examples, there is no need to calculate all degrees. It is enough simply to reduce the degrees with the same indicators and bases.
In order to finally grasp how to reduce fractions with degrees, one must practice a lot. After several similar examples, the actions will be performed automatically.
And if the expression is the root?
It can also be shortened. Only again, following the rules. And all those that were described above are true. In general, if there is a question about how to reduce the fraction with roots, then it is necessary to divide.
Irrational expressions can also be divided. That is, if the numerator and denominator have the same multipliers, concluded under the root sign, then they can be boldly reduced. This will simplify the expression and complete the job.
If, after the contraction under the bar, the irrationality remains, then you need to get rid of it. In other words, multiply the numerator and denominator. If after this operation there are common multipliers, then they again need to be reduced.
Here, perhaps, and all about how to reduce fractions. There were few rules, but one ban. Never shorten the summands!