Square expressions. Quadratic equations

In this lesson, we will study the basic property of a fraction, find out which fractions are equal to each other. We will learn how to reduce fractions, determine whether a fraction is reduced or not, practice reducing fractions and find out when to use reduction and when not.

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Basic property of a fraction

Imagine such a situation.

At the table 3 human and 5 apples. Divide 5 three apples. Each gets \(\mathbf(\frac(5)(3))\) apples.

And at the next table 3 person and also 5 apples. Each again \(\mathbf(\frac(5)(3))\)

At the same time, all 10 apples 6 Human. Each \(\mathbf(\frac(10)(6))\)

But it's the same.

\(\mathbf(\frac(5)(3) = \frac(10)(6))\)

These fractions are equivalent.

You can double the number of people and double the number of apples. The result will be the same.

In mathematics, this is formulated as follows:

If the numerator and denominator of a fraction are multiplied or divided by the same number (not equal to 0), then the new fraction will be equal to the original.

This property is sometimes referred to as " basic property of a fraction ».

$$\mathbf(\frac(a)(b) = \frac(a\cdot c)(b\cdot c) = \frac(a:d)(b:d))$$

For example, the way from the city to the village- 14 km.

We walk along the road and determine the distance traveled by the kilometer posts. After passing six columns, six kilometers, we understand that we have passed \(\mathbf(\frac(6)(14))\) paths.

But if we do not see the poles (maybe they have not been installed), we can count the path along the electric poles along the road. Them 40 pieces per kilometer. That is, everything 560 all the way. Six kilometers - \(\mathbf(6\cdot40 = 240)\) pillars. That is, we passed 240 from 560 columns- \(\mathbf(\frac(240)(560))\)

\(\mathbf(\frac(6)(14) = \frac(240)(560))\)

Example 1

Mark a point with coordinates ( 5; 7 ) on the coordinate plane XOY. It will match the fraction \(\mathbf(\frac(5)(7))\)

Connect the origin to the resulting point. Construct another point that has coordinates twice the previous ones. What fraction did you get? Will they be equal?

Decision

A fraction on the coordinate plane can be marked with a dot. To draw a fraction \(\mathbf(\frac(5)(7))\), mark a point with coordinate 5 along the axis Y and 7 along the axis X. Let's draw a straight line from the origin through our point.

The point corresponding to the fraction \(\mathbf(\frac(10)(14))\)

They are equivalent: \(\mathbf(\frac(5)(7) = \frac(10)(14))\)

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 in higher mathematics. If the student has not learned, then he may have problems in algebra. Therefore, it is better to understand a few rules once and for all. And remember one prohibition and never break it.

Fraction and its reduction

What it is, every student knows. Any two digits located between the horizontal bar are immediately perceived as a fraction. However, not everyone understands that any number can become it. If it is an integer, then it can always be divided by one, then you get an improper fraction. But more on 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 less than the denominator. To do this, you need to remember the main property of a fraction. It states that when multiplying (as well as dividing) both its numerator and denominator by the same number turns out to be the same as the original fraction.

The division actions that are performed on this property result in a reduction. That is, its maximum simplification. A fraction can be reduced as long as there are common factors above and below the line. When they no longer exist, the reduction is impossible. And they say that this fraction is irreducible.

two ways

1.Step by step reduction. It uses the guessing method, when both numbers are divided by the minimum common factor that the student noticed. If after the first reduction it is clear 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 means decomposing the numerator and denominator into prime factors. Among them, then you need to choose all the same. Their product will give the largest common factor by which the fraction is reduced.

Both of these methods are equivalent. The student is invited to master them and use the one that he liked best.

What if there are letters and operations of addition and subtraction?

With the first part of the question, everything is more or less clear. Letters can be abbreviated just like numbers. The main thing is that they act as multipliers. But with the second, many have problems.

Important to remember! You can only reduce numbers that are factors. If they are terms, it is impossible.

In order to understand how to reduce fractions that look like algebraic expression, you need to learn the rule. First, express the numerator and denominator as a product. Then you can reduce if there are common factors. For representation as multipliers, the following tricks are useful:

• grouping;
• bracketing;
• application of abbreviated multiplication identities.

Moreover, the latter method makes it possible to immediately obtain terms in the form of factors. Therefore, it must always be used if a known pattern is visible.

But this is not scary yet, then tasks with degrees and roots appear. That's when you need to muster up the courage and learn a couple of new rules.

Power expression

Fraction. The product in the numerator and denominator. 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 powers, you need to learn two points:

• if there is a sum in the exponent, then it can be decomposed into factors, the powers of which will be the original terms;
• if the difference, then into the dividend and the divisor, the first in the degree will be reduced, the second - subtracted.

After completing these steps, the common multipliers become visible. In such examples, it is not necessary to calculate all powers. It is enough to simply reduce the degrees with the same indicators and bases.

In order to finally master how to reduce fractions with powers, you need a lot of practice. After several examples of the same type, the actions will be performed automatically.

What if the expression contains a root?

It can also be shortened. Again, just follow the rules. Moreover, all those described above are true. In general, if the question is how to reduce a fraction with roots, then you need to divide.

On the irrational expressions can also be divided. That is, if the numerator and denominator are same multipliers enclosed under the sign of the root, then they can be safely reduced. This will simplify the expression and get the job done.

If, after the reduction, irrationality remains under the line of the fraction, then you need to get rid of it. In other words, multiply the numerator and denominator by it. If after this operation common factors appeared, then they will need to be reduced again.

That, perhaps, is all about how to reduce fractions. Few rules, but one prohibition. Never reduce terms!

Calculator online performs reduction of algebraic fractions according to the rule of reduction of fractions: replacement of the original fraction equal fraction, but with smaller numerator and denominator, i.e. simultaneous division of the numerator and denominator of a fraction by their common largest common divisor(GCD). The calculator also displays detailed solution, which will help you understand the sequence of execution of the reduction.

Given:

Decision:

Doing Fraction Reduction

checking the possibility of performing a reduction algebraic fraction

1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction

determination of the greatest common divisor (gcd) of the numerator and denominator of an algebraic fraction

2) Reducing the numerator and denominator of a fraction

reduction of the numerator and denominator of an algebraic fraction

3) Selection of the integer part of the fraction

extracting the integer part of an algebraic fraction

4) Converting an algebraic fraction to a decimal fraction

conversion of algebraic fraction to decimal

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I. The procedure for reducing an algebraic fraction with an online calculator:

1. To reduce an algebraic fraction, enter the values ​​of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the integer part of the fraction. If the fraction is simple, then leave the integer part field blank.
2. To set negative fraction, put a minus sign in the integer part of the fraction.
3. Depending on the given algebraic fraction, the following sequence of actions is automatically performed:

• determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
• reduction of the numerator and denominator of a fraction by gcd;
• extracting the integer part of a fraction if the numerator of the final fraction is greater than the denominator.
• converting the final algebraic fraction to a decimal fraction rounded to hundredths.

The result of the reduction may be an improper fraction. In this case, the final proper fraction will be allocated whole part and the resulting fraction will be converted to a proper fraction.

II. For reference:

A fraction is a number consisting of one or more parts (fractions) of a unit. An ordinary fraction (simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction), separated by a horizontal bar (fractional bar), denoting the sign of division. The numerator of a fraction is the number above the fraction bar. The numerator shows how many parts were taken from the whole. The denominator of a fraction is the number below the fractional bar. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have an integer part. A simple fraction can be right or wrong. A proper fraction is a fraction whose numerator less than the denominator, so a proper fraction is always less than one. Example of correct fractions: 8/7, 11/19, 16/17. An improper fraction is a fraction whose numerator is greater than or equal to the denominator, so an improper fraction is always greater than or equal to one. Example improper fractions: 7/6, 8/7, 13/13. mixed fraction - a number that includes an integer and a proper fraction, and denotes the sum of this integer and a proper fraction. Any mixed fraction can be converted to an improper simple fraction. Example mixed fractions: 1¼, 2½, 4¾.

III. Note:

1. Source data block highlighted yellow , block of intermediate calculations highlighted blue color , solution block highlighted in green.
2. For addition, subtraction, multiplication and division of ordinary or mixed fractions, use the online fraction calculator with a detailed solution.

We will understand what fraction reduction is, why and how to reduce fractions, we will give the rule for reducing fractions and examples of its use.

Yandex.RTB R-A-339285-1

What is "fraction reduction"

Reduce fraction

To reduce a fraction means to divide its numerator and denominator by a common divisor, positive and different from one.

As a result of such an action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.

For example, let's take common fraction 6 24 and shorten it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12 . In this example, we have reduced the original fraction by 2 .

Reduction of fractions to irreducible form

In the previous example, we reduced the fraction 6 24 by 2 , resulting in the fraction 3 12 . It is easy to see that this fraction can be further reduced. Generally, the goal of reducing fractions is to end up with an irreducible fraction. How to convert a fraction to an irreducible form?

This can be done by reducing the numerator and denominator by their greatest common divisor (GCD). Then, by the property of the greatest common divisor, in the numerator and in the denominator there will be mutually prime numbers, and the fraction is irreducible.

a b = a ÷ N O D (a , b) b ÷ N O D (a , b)

Reduction of a fraction to an irreducible form

To reduce a fraction to an irreducible form, you need to divide its numerator and denominator by their gcd.

Let's return to the fraction 6 24 from the first example and reduce it to an irreducible form. The greatest common divisor of 6 and 24 is 6 . Let's reduce the fraction:

6 24 = 6 ÷ 6 24 ÷ 6 = 1 4

Reducing fractions is convenient to use so as not to work with large numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. By reducing a fraction, most often they mean its reduction to an irreducible form, and not just reduction by a common divisor of the numerator and denominator.

Fraction reduction rule

To reduce fractions, it is enough to remember the rule, which consists of two steps.

Fraction reduction rule

To reduce a fraction:

1. Find the gcd of the numerator and denominator.
2. Divide the numerator and denominator by their gcd.

Consider practical examples.

Example 1. Let's reduce the fraction.

Given a fraction 182 195 . Let's shorten it.

Find the GCD of the numerator and denominator. For this in this case The best way is to use Euclid's algorithm.

195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13

Divide the numerator and denominator by 13. We get:

182 195 = 182 ÷ 13 195 ÷ 13 = 14 15

Ready. We got an irreducible fraction, which is equal to the original fraction.

How else can you reduce fractions? In some cases it is convenient to decompose the numerator and denominator into simple factors, and then from the upper and lower parts fractions to remove all common factors.

Example 2. Reduce the fraction

Given a fraction 360 2940 . Let's shorten it.

To do this, we represent the original fraction in the form:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7

Let's get rid of the common factors in the numerator and denominator, as a result of which we get:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49

Finally, consider another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, at each of which the fraction is reduced by some obvious common divisor.

Example 3. Reduce the fraction

Let's reduce the fraction 2000 4400 .

It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:

2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44

20 44 = 20 ÷ 2 44 ÷ 2 = 10 22

The resulting result is again reduced by 2 and we get an irreducible fraction:

10 22 = 10 ÷ 2 22 ÷ 2 = 5 11

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