Fractions and their reduction is another topic that starts in 5th grade. Here the base of this action is formed, and then these skills are pulled by a thread to 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 a few rules once and for all. And remember one ban and never violate it.
Fraction and its reduction
What it is, every student knows. Any two digits 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 by one, then you get an incorrect fraction. But more about that later.
The beginning is always simple. First you need to figure out how to reduce the correct fraction. That is, one whose numerator is smaller than the denominator. For this you need to remember the main property of the fraction. It asserts that by multiplying (as well as dividing) its numerator and denominator by the same number at the same time, it is equivalent to the original fraction.
Division actions that are performed in this property and lead to a reduction. That is, its maximum simplification. Fraction can be reduced as long as there are common factors above and below the line. When they are gone, the reduction is impossible. And they say that this fraction is irreducible.
1. Step by step reduction. It uses the method of estimation, when both numbers are divided by the minimum common factor that the student noticed. If after the first contraction it can be seen that this is not the end, then the division continues. Until the fraction becomes irreducible.
2. Finding the greatest common divisor of the numerator and denominator. This is the most rational way to reduce fractions. It implies a decomposition of the numerator and denominator into prime factors. Among them, then you need to choose all the same. Their product will give the greatest common factor, which is reduced fraction.
Both of these methods are equivalent. The student is invited to master them and use the one that they liked the most.
What if there are letters and actions of addition and subtraction?
The first part of the question is more or less clear. Letters can be abbreviated as numbers. The main thing is that they act as multipliers. But with the second, many have problems.
It is important to remember! You can reduce only the numbers that are multipliers. If they are terms, 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:
- imposing a bracket;
- application of identities of abbreviated multiplication.
Moreover, the latter method makes it possible to immediately obtain the terms in the form of multipliers. Therefore, it must always be used if a known pattern is visible.
But it is still not scary, then tasks with degrees and roots appear. That's when you need courage and learn a couple of new rules.
Expression with degree
Fraction. In the numerator and denominator product. There are letters and numbers. And they are also raised to a power, which also consists of terms or factors. There is something to be afraid of.
In order to figure out how to reduce fractions with degrees, you need to learn two things:
- if the exponent is a sum, then it can be expanded into factors, the degrees of which will be the original terms;
- if the difference is, then the dividend and the divisor, the first will have a decrease in the degree, the second - a deductible.
After performing these actions, the common factors become visible. In such examples, it is not necessary to calculate all degrees. It is enough just to reduce the degrees with the same indicators and bases.
In order to finally learn how to reduce fractions with degrees, you need to practice a lot. After several examples of the same type, actions will be performed automatically.
And if the expression is the root?
It can also be reduced. But 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 the roots, then you need to divide.
Irrational expressions can also be divided. That is, if the numerator and denominator are identical factors enclosed under the sign of the root, then they can be safely reduced. This will simplify the expression and perform the task.
If, after the reduction under the fraction line, there remains irrationality, then it is necessary to get rid of it. In other words, multiply the numerator and denominator. If after this operation there are common factors, they will need to be reduced again.
Here, perhaps, all about how to reduce fractions. There are few rules, but one prohibition. Never shorten terms!