Obviously, numbers with powers can be added like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2 .
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4 .

Odds the same powers of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2 .

It is also obvious that if we take two squares a, or three squares a, or five squares a.

But degrees various variables and various degrees identical variables, must be added by adding them to their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3 .

It is obvious that the square of a, and the cube of a, is neither twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6 .

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahend must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Power multiplication

Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3 .

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n is;

And a m , is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are - negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y-n .y-m = y-n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y).(a + y) = a 2 - y 2 .
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4 .
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8 .

Division of degrees

Numbers with powers can be divided like other numbers by subtracting from the divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is a 3 .

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing powers with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master the multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Reduce the exponents in $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.

) and the denominator by the denominator (we get the denominator of the product).

Fraction multiplication formula:

For example:

Before proceeding with the multiplication of numerators and denominators, it is necessary to check for the possibility of fraction reduction. If you manage to reduce the fraction, then it will be easier for you to continue to make calculations.

Division of an ordinary fraction by a fraction.

Division of fractions involving a natural number.

It's not as scary as it seems. As in the case of addition, we convert an integer into a fraction with a unit in the denominator. For example:

Multiplication of mixed fractions.

Rules for multiplying fractions (mixed):

• convert mixed fractions to improper;
• multiply the numerators and denominators of fractions;
• we reduce the fraction;
• if we get an improper fraction, then we convert the improper fraction to a mixed one.

Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.

The second way to multiply a fraction by a natural number.

It is more convenient to use the second method of multiplying an ordinary fraction by a number.

Note! To multiply a fraction by a natural number, it is necessary to divide the denominator of the fraction by this number, and leave the numerator unchanged.

From the above example, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.

Multilevel fractions.

In high school, three-story (or more) fractions are often found. Example:

To bring such a fraction to its usual form, division through 2 points is used:

Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.

Note, for example:

When dividing one by any fraction, the result will be the same fraction, only inverted:

Practical tips for multiplying and dividing fractions:

1. The most important thing in working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It is better to write down a few extra lines in a draft than to get confused in the calculations in your head.

2. In tasks with different types of fractions - go to the type of ordinary fractions.

3. We reduce all fractions until it is no longer possible to reduce.

4. We bring multi-level fractional expressions into ordinary ones, using division through 2 points.

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

In the last lesson, we learned how to add and subtract decimal fractions (see the lesson " Adding and subtracting decimal fractions"). At the same time, they estimated how much the calculations are simplified compared to the usual “two-story” fractions.

Unfortunately, with multiplication and division of decimal fractions, this effect does not occur. In some cases, decimal notation even complicates these operations.

First, let's introduce a new definition. We will meet him quite often, and not only in this lesson.

The significant part of a number is everything between the first and last non-zero digit, including the trailers. We are only talking about numbers, the decimal point is not taken into account.

The digits included in the significant part of the number are called significant digits. They can be repeated and even be equal to zero.

For example, consider several decimal fractions and write out their corresponding significant parts:

1. 91.25 → 9125 (significant figures: 9; 1; 2; 5);
2. 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
3. 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
4. 0.0304 → 304 (significant figures: 3; 0; 4);
5. 3000 → 3 (there is only one significant figure: 3).

Please note: zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary ones (see the lesson “ Decimal Fractions”).

This point is so important, and errors are made here so often that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of a significant part, will proceed, in fact, to the topic of the lesson.

Decimal multiplication

The multiplication operation consists of three consecutive steps:

1. For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
2. Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We get the significant part of the desired fraction;
3. Find out where and by how many digits the decimal point is shifted in the original fractions to obtain the corresponding significant part. Perform reverse shifts on the significant part obtained in the previous step.

Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.

1. 0.28 12.5;
2. 6.3 1.08;
3. 132.5 0.0034;
4. 0.0108 1600.5;
5. 5.25 10,000.

We work with the first expression: 0.28 12.5.

1. Let's write out the significant parts for the numbers from this expression: 28 and 125;
2. Their product: 28 125 = 3500;
3. In the first multiplier, the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second - by another 1 digit. In total, a shift to the left by three digits is needed: 3500 → 3.500 = 3.5.

Now let's deal with the expression 6.3 1.08.

1. Let's write out the significant parts: 63 and 108;
2. Their product: 63 108 = 6804;
3. Again, two shifts to the right: by 2 and 1 digits, respectively. In total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no zeros at the end.

We got to the third expression: 132.5 0.0034.

1. Significant parts: 1325 and 34;
2. Their product: 1325 34 = 45,050;
3. In the first fraction, the decimal point goes to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We perform a shift by 5 to the left: 45050 → .45050 = 0.4505. Zero was removed at the end, and added to the front so as not to leave a “bare” decimal point.

The following expression: 0.0108 1600.5.

1. We write significant parts: 108 and 16 005;
2. We multiply them: 108 16 005 = 1 728 540;
3. We count the numbers after the decimal point: in the first number there are 4, in the second - 1. In total - again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.

Finally, the last expression: 5.25 10,000.

1. Significant parts: 525 and 1;
2. We multiply them: 525 1 = 525;
3. The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52 500 (we had to add zeros).

Pay attention to the last example: since the decimal point moves in different directions, the total shift is through the difference. This is a very important point! Here's another example:

Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12 500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 digits to the left. As a result, we stepped 2 − 1 = 1 digit to the left.

Decimal division

Division is perhaps the most difficult operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case, there are many subtleties that negate the potential savings.

So let's look at a generic algorithm that is a little longer, but much more reliable:

1. Convert all decimals to common fractions. With a little practice, this step will take you a matter of seconds;
2. Divide the resulting fractions in the classical way. In other words, multiply the first fraction by the "inverted" second (see the lesson " Multiplication and division of numerical fractions");
3. If possible, return the result as a decimal. This step is also fast, because often the denominator already has a power of ten.

A task. Find the value of the expression:

1. 3,51: 3,9;
2. 1,47: 2,1;
3. 6,4: 25,6:
4. 0,0425: 2,5;
5. 0,25: 0,002.

We consider the first expression. First, let's convert obi fractions to decimals:

We do the same with the second expression. The numerator of the first fraction is again decomposed into factors:

There is an important point in the third and fourth examples: after getting rid of the decimal notation, cancellable fractions appear. However, we will not perform this reduction.

The last example is interesting because the numerator of the second fraction is a prime number. There is simply nothing to factorize here, so we consider it “blank through”:

Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.

In addition, when dividing, “ugly” fractions often appear that cannot be converted to decimals. This is where division differs from multiplication, where the results are always expressed in decimal form. Of course, in this case, the last step is again not performed.

Pay also attention to the 3rd and 4th examples. In them, we deliberately do not reduce ordinary fractions obtained from decimals. Otherwise, it will complicate the inverse problem - representing the final answer again in decimal form.

Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.

Example.

Find the product of algebraic fractions and.

Solution.

Before performing the multiplication of fractions, we factorize the polynomial in the numerator of the first fraction and the denominator of the second. The corresponding abbreviated multiplication formulas will help us with this: x 2 +2 x+1=(x+1) 2 and x 2 −1=(x−1) (x+1) . In this way, .

Obviously, the resulting fraction can be reduced (we discussed this process in the article on the reduction of algebraic fractions).

It remains only to write the result in the form of an algebraic fraction, for which you need to multiply the monomial by the polynomial in the denominator: .

Usually, the solution is written without explanation as a sequence of equalities:

Answer:

.

Sometimes with algebraic fractions that need to be multiplied or divided, some transformations should be performed to make the implementation of these operations easier and faster.

Example.

Divide an algebraic fraction by a fraction.

Solution.

Let's simplify the form of an algebraic fraction by getting rid of the fractional coefficient. To do this, we multiply its numerator and denominator by 7, which allows us to make the main property of an algebraic fraction, we have .

Now it has become clear that the denominator of the resulting fraction and the denominator of the fraction by which we need to divide are opposite expressions. Change the signs of the numerator and denominator of the fraction , we have .

Pure mathematics is in its way the poetry of the logical idea. Albert Einstein

In this article, we offer you a selection of simple mathematical tricks, many of which are quite relevant in life and allow you to count faster.

1. Fast interest calculation

Perhaps, in the era of loans and installments, the most relevant mathematical skill can be called a virtuoso mental calculation of interest. The fastest way to calculate a certain percentage of a number is to multiply the given percentage by this number and then discard the last two digits in the resulting result, because the percentage is nothing but one hundredth.

How much is 20% of 70? 70 × 20 = 1400. We discard two digits and get 14. When you rearrange the factors, the product does not change, and if you try to calculate 70% of 20, then the answer will also be 14.

This method is very simple in the case of round numbers, but what if you need to calculate, for example, a percentage of the number 72 or 29? In such a situation, you will have to sacrifice accuracy for the sake of speed and round the number (in our example, 72 is rounded up to 70, and 29 to 30), and then use the same trick with multiplying and discarding the last two digits.

2. Quick divisibility check

Can 408 candies be divided equally between 12 children? It is easy to answer this question without the help of a calculator, if we recall the simple signs of divisibility that we were taught back in school.

• A number is divisible by 2 if its last digit is divisible by 2.
• A number is divisible by 3 if the sum of the digits that make up the number is divisible by 3. For example, take the number 501, represent it as 5 + 0 + 1 = 6. 6 is divisible by 3, which means that the number 501 itself is divisible by 3 .
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, take 2340. The last two digits form the number 40, which is divisible by 4.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 6 if it is divisible by 2 and 3.
• A number is divisible by 9 if the sum of the digits that make up the number is divisible by 9. For example, let's take the number 6,390 and represent it as 6 + 3 + 9 + 0 = 18. 18 is divisible by 9, which means the number 6 itself 390 is divisible by 9.
• A number is divisible by 12 if it is divisible by 3 and 4.

3. Fast calculation of the square root

The square root of 4 is 2. Anyone can count that. What about the square root of 85?

For a quick approximate solution, we find the nearest square number to the given one, in this case it is 81 = 9^2.

Now find the next nearest square. In this case it is 100 = 10^2.

The square root of 85 is somewhere between 9 and 10, and since 85 is closer to 81 than it is to 100, the square root of that number is 9 something.

4. Quick calculation of the time after which a cash deposit at a certain percentage will double

Do you want to quickly find out the time it will take for your cash deposit at a certain interest rate to double? There is also no need for a calculator, it is enough to know the “rule of 72”.

We divide the number 72 by our interest rate, after which we get the approximate period after which the deposit will double.

If the deposit is made at 5% per annum, then it will take 14-odd years for it to double.

Why exactly 72 (sometimes they take 70 or 69)? How it works? These questions will be answered in detail by Wikipedia.

5. Quick calculation of the time after which a cash deposit at a certain percentage will triple

In this case, the interest rate on the deposit should become a divisor of 115.

If the deposit is made at 5% per annum, then it will take 23 years for it to triple.

6. Quick calculation of the hourly rate

Imagine that you are interviewing with two employers who do not state salaries in the usual “rubles per month” format, but talk about annual salaries and hourly pay. How to quickly calculate where they pay more? Where the annual salary is 360,000 rubles, or where they pay 200 rubles per hour?

To calculate the payment for one hour of work when voicing the annual salary, it is necessary to discard the last three characters from the named amount, and then divide the resulting number by 2.

360,000 turns into 360 ÷ 2 = 180 rubles per hour. Other things being equal, it turns out that the second proposal is better.

7. Advanced math on fingers

Your fingers are capable of much more than simple addition and subtraction.

With your fingers, you can easily multiply by 9 if you suddenly forgot the multiplication table.

Let's number the fingers on the hands from left to right from 1 to 10.

If we want to multiply 9 by 5, then we bend the fifth finger from the left.

Now let's look at the hands. It turns out four unbent fingers to bent. They represent tens. And five unbent fingers after the bent one. They represent units. Answer: 45.

If we want to multiply 9 by 6, then we bend the sixth finger from the left. We get five unbent fingers before the bent finger and four after. Answer: 54.

Thus, you can reproduce the entire column of multiplication by 9.

8. Fast multiplication by 4

There is an extremely easy way to lightning-fast multiply even large numbers by 4. To do this, it is enough to decompose the operation into two steps, multiplying the desired number by 2, and then again by 2.

See for yourself. Not everyone can multiply 1,223 immediately by 4 in their minds. And now we do 1223 × 2 = 2446 and then 2446 × 2 = 4892. This is much easier.

9. Quick determination of the required minimum

Imagine that you are taking a series of five tests, for which you need a minimum score of 92 to pass. The last test remains, and the results for the previous ones are: 81, 98, 90, 93. How to calculate the required minimum that you need to get in the last test?

To do this, we consider how many points we missed / went over in the tests already passed, denoting the shortage with negative numbers, and the results with a margin - positive.

So, 81 − 92 = −11; 98 - 92 = 6; 90 - 92 = -2; 93 - 92 = 1.

Adding these numbers, we get the adjustment for the required minimum: -11 + 6 - 2 + 1 = -6.

It turns out a deficit of 6 points, which means that the required minimum increases: 92 + 6 = 98. Things are bad. :(

10. Quick representation of the value of an ordinary fraction

The approximate value of an ordinary fraction can be very quickly represented as a decimal fraction, if you first bring it to simple and understandable ratios: 1/4, 1/3, 1/2 and 3/4.

For example, we have a fraction 28/77, which is very close to 28/84 = 1/3, but since we increased the denominator, the original number will be slightly larger, that is, slightly more than 0.33.

11. Number Guessing Trick

You can play a bit of David Blaine and surprise your friends with an interesting but very simple math trick.

1. Ask a friend to guess any whole number.
2. Let him multiply it by 2.
3. Then add 9 to the resulting number.
4. Now let's subtract 3 from the resulting number.
5. And now let him divide the resulting number in half (it will be divided without a remainder anyway).
6. Finally, ask him to subtract from the resulting number the number that he thought of at the beginning.

The answer will always be 3.

Yes, very stupid, but often the effect exceeds all expectations.

Bonus

And, of course, we could not help but insert into this post that same picture with a very cool way of multiplying.