Previously, we advise you, as you read, to solve the exercises mentioned in examples. Each completed exercise, compare your result to the one given to you. A few days later, repeat this time without looking at the development, but always compare the results.
We repeat it, in mathematics, it is necessary to practice and to repeat to understand and to retain.
Pursuing. If we consider, as we have said, that the four elementary operations (addition, subtraction, multiplication, division) are known to the reader, it seems useful to us to speak of "the rise to a power" and its inverse, that is, "extraction of a root". The six arithmetic operations will then be known.
Raising to a power is a special form of multiplication.
Definition:The power of a number is the product of several equal numbers between them..
For example, the number 100, which can be obtained by multiplying 10 by 10, is a power of 10. The number 1000, which can be obtained by multiplying 10 by 10 and the result by 10, is another power of 10. The number 16, which can be obtained by multiplying 2 by 2, the result by 2 and the new result still by 2, is a power of 2. What is written : 10. 10 = 100, 10. 10. 10 = 1000, 2. 2. 2. 2 = 16.
However, in ordinary mathematical expressions, to indicate these calculations, we have recourse to a convention which has the advantage of being very concise and of highlighting the properties of the powers.
Since the factors of multiplication used for the calculation of a power are all equal, the number serving as a factor is written once and then the number representing the quantity is written next to it, at the top right. of factors. By doing so, the power rise of the preceding examples will be written as follows :
102 = 100 ; 103 = 1 000 ; 24 = 16
The number that represents the equal factors is called the base (the numbers 10, 10 and 2 are the bases of the respective powers taken into consideration). The other number, which indicates the quantity of these same factors is called the exponent (the numbers 2, 3 and 4, given examples are exponents).
The base and exponent set forms the power.
When the exponent is 2, the power is called the square or power two. When the exponent is 3, the power is called the cube or power three. With the exhibitors 4, 5, 6 ... They say respectively power four, power five, power six, etc. ...
In relation to these powers, the operations used to calculate their values are respectively called squaring, either at the power of two, elevation at the cube, or at the power of three, elevation at the power of four, at the power of five. , to the power six, etc ... Note that this terminology used in everyday language is not rigorous because we should say : 10 exponent 2, 10 exponent 3, 2 exponent 4 instead of 10 power 2, 10 power 3 , 2 power 4, etc ...
It can be very useful to write a number in the form of elevation to a power to get abbreviated arithmetic expressions, especially when it comes to very large numbers made up of few digits and followed by many zeros, or in the calculation of very small numbers, consisting of decimal digits preceded by many zeros.
To fix the ideas, consider the case of a very large number : one hundred billion.
In the ordinary form, this number is written with eleven zeros after the number 1, or 100 000 000 000.
Obviously, it is a little tedious to write such a large number; however, if we consider that :
and so that it is a power of 10, we can after counting the number of factors (which is 11, just like the number of zeros), write more concisely 1011, instead of aligning a long line of zeros.
Similarly, considering that the number :
1 200 000 is equal to
12 . 10 . 10 . 10 . 10 . 10,
We can transform in elevation to a power the part of the multiplication which follows the 12. The result will be :
1 200 000 = 12 . 105
The expression of numbers in the form of elevation at a power, which in a very obvious way is very useful for shortening the writing, can also lead to a noticeable simplification of arithmetic operations, when one has to perform calculations with two or more powers.
Let's see briefly the main rules to which the calculations of the powers are obeyed.
Rule 1:The product of two or more having the same base, is equal to a power having the same base and an exponent equal to the sum of the exponents.
Examples:
102 . 103
. 106 = 102 + 3 + 6 = 1011
53 .
52 . 51 = 53 + 2 + 1 = 56
Rule 2:The division of two powers having the same base is equal to a power having the same base and an exponent equal to the difference of the exponents.
Example :
74 / 72 = 74-2 = 72
In the applications of this rule, we can meet three particular cases that we will examine.
CAS 1:the difference of the exponents is equal to 1.
Example:
34 / 33 = 34-3 = 31
The exponent of the power which constitutes the result of the operation is equal to 1. In this particular case, we thus have 31 = 3, as previously we had 51 = 5. Therefore, the exponent 1 means that it There is only one factor.
CAS 2:the difference of the exponents is equal to zero.
Example:
112 / 112
= 112 - 2
= 110
Since
112 = 121, we have :
112 / 112 = 121 / 121 = 1
therefore : 112
/ 112 = 110 = 121 /121 = 1
The value that must be assigned to the power 110 is therefore 1.
In the same way, we find, for example :
103 / 103
= 100 = 1 000 / 1 000 = 1
0,52 / 0,52
= 0,50 = 0,5 x 0,5 / 0,5 x 0,5 = 0,25 / 0,25 = 1
It is interesting to note that the powers with an exponent equal to zero always have the same value : they are equal to 1, so we will have :
20 = 1 ;
250 = 1 ; 10 0000 = 1
and so on for any other base with zero exponent.
CAS 3:the difference of the exponents is equal to a negative number.
The value of the power having a negative exponent, 7-3
is therefore equal to 1 / 343, is 1 / 73
Consequently, any power having a negative exponent is equal to a fraction having as numerator 1 and denominator that same power whose exponent has been made positive.
One can thus make positive all the negative exponents, for example :
Rule 3:- The product of two or more powers having the same exponent is equal to a power of the same exponent based on the product of the bases.
102 x 52 = (10 x 5)2 = 502 = 2 500
Rule 4:-
The division of two powers having the same exponent is equal to a power whose exponent is the same and whose base is equal to the quotient of the two bases.
Root extraction is the reverse operation from elevation to power. As there are powers two, three, four ... There are roots second or square, third or cubic, fourth ... In practice, it is the square root that we meet most often. It is therefore she who will be more particularly treated in the following lines.
4. 1. - SQUARE ROOT
Definition :The square root of a number A is the number B which, multiplied twice by itself, will be equal to A.
Definition:The cubic root of a number A is the number B which, multiplied three times by itself, will be equal to A.
Examples:
The cubic root of 8 is 2 because 2 x 2 x 2 = 8
The cubic root of 27 is 3 because 3 x 3 x 3 = 27
The cubic root of 64 is 4 because 4 x 4 x 4 = 64
4. 3. - ROOT FOURTH, FIFTH
Definition:The fourth (fifth ...) root of a number A is the number B which, multiplied four times (five times ...) by itself, will be equal to A.
Examples:
Root fourth of 16 is 2 because 2 x 2 x 2 x 2 = 16
Root fifth of 243 is 3 because 3 x 3 x 3 x 3 x 3 = 243
4. 4. - EXTRACTION OF THE SQUARE ROOT
We will now study how to extract a square root. This operation requires a very detailed explanation. But when you have understood the procedure, you will see that the extraction of a square root does not require much more time than a division.
We will illustrate our explanations with an example ; Find the square root of 266,0161 :
Process:
1 - We write the number under the radical (sign of the operation of extracting a root).
As has been said, the index 2, which should distinguish the square root of higher index roots (3, 4, 5 ...) is omitted.
2 - Separate the given number "in slices" of two digits starting from the right to the left, or, if it is like in our example of a decimal number, starting on the right and on the left of the comma.
2 . 66,01 . 61
It may happen that the last group on the left is a single digit, like the 2 in our example. The same thing could happen to the last group to the right of the comma ; in this case, always add a zero, so that after the comma all groups are formed of two digits.
3 - Let us examine the first group on the left, namely 2. We calculate mentally what is the integer which, when squared, makes it possible to obtain the result nearest to, equal to or lower than, the number formed by the first group of figures.
In our case, this integer is 1, since the square of 1, that is, 1 x 1 is equal to 1 (while the next integer, 2, squared, gives the result 4, which is bigger than 2). We write the number found in the space reserved for the root.
If the number formed by the digits in the first group was 21, the searched integer would be four, because the square 4 x 4 = 16 is less than 21, while the square of the next whole number, 5 x 5 = 25, If, however, the number of the first group was 25, the whole number sought would be 5, because 5 x 5 = 25.
4 - The square of the integer previously found under the number of the first group is plotted and the subtraction is carried out. Since the square of 1 is equal to 1, under the number 2 we write 1 and calculate the difference.
5 - Next to the difference found, the figures of the second group, 66, are reported, and the last digit is separated by a point.
6 - We double the number in the space reserved for the root (1 x 2 = 2) and we write below the horizontal line the product obtained.
7 - One calculates mentally the quotient between the number 16, obtained by the separation of the last digit of the number 166, and the number 2, obtained by the doubling of the present digit, in the space reserved for the roots.
Then we write the quotient next to the 2, under the space of the roots. Finally, we multiply the result number by the same quotient. Since 16 / 2 = 8, next to 2 we write this number and multiply the resulting number (28) by 8.
8 - The product 224 is compared with the number 166.
The product is larger than 166. So repeat the previous operations by decreasing the quotient by one unit, that is, using the number 7 instead of the number 8. However, even with 7, the product (189) is greater than 166. Repeat the same operations with the number less than 7 or with 6.
The product is less than 166. We stop this series of operations by drawing a horizontal line under the last multiplication.
9 - The product 156 is written under the number 166 and the subtraction is carried out. In addition, next to the difference, the third group of digits, namely 01, is reported by separating the last digit by a dot.
10 - We report 6 (that is, the number that allowed us to get the product 156 smaller than 166) in the root space, next to the number 1.
The two digits form the number 16. The number 16 must be doubled and the new product (16 x 2 = 32) must be reported under the previous multiplications.
After the number 6, in the space reserved for the root, we put the comma, since the operation is completed for the integer part of the number 266, 0161.
11 - Now, by repeating again the process started in point 7 (see above), we look for the quotient of the division 100 / 32, by only postponing the integer part of the quotient. Then we write next to 32 the number that represents the integer part of the quotient, namely 3 (100 / 32 = 3, ...), and we do the multiplication by 3.
Since the product obtained (969) is less than 1001, the subtraction between the two numbers is immediately carried out. In addition, next to the difference, the last group of digits, 61, is reported by separating the last digit by one point.
12 - We postpone the number 3 (that is, the number that allowed us to get the 969 product, smaller than 1 001) in the root space immediately after the decimal point. Then, we double the number 163, obtained by adding 3 after 16 and omitting the comma, and we write the product (163 x 2 = 326) under the last multiplication.
13 - Get to this point, let's look at the number 326 obtained by separating the last digit from 3 261 and the other number 326, obtained by doubling 16,3 and deleting the comma.
The division of these two numbers 326 / 326 is equal to 1. Therefore, following the method described in points 7 and 11, 1 to 326 is added and the number thus formed is multiplied by 1. The result of this multiplication is subtracted from 3 261. Since the product is also equal to 3 261, the difference will be zero.
We write the number 1, with which we performed the previous operations, next to the last digit of the number 16,3 in the space reserved for the root.
The calculation is finished (OUF !). Result 16,31 is the exact square root of the number 266,0161 since the remainder is zero.
To verify this result, simply calculate the square of the root.
In our case, by multiplying 16.31 x 16.31, we obtain 266.0161, which is precisely the number whose root we calculated. We can therefore conclude that the result does not contain an error.
To practice, let's perform, without describing all the operations, the calculation of the square root of the number 179.
The result is 13, but since the operation has a remainder, 10, the number 13 is not the exact square root of 179.
In such cases, the operation can be continued to find other decimals which, added to the whole root, form a number as close as possible to the exact value of the root.
To this end, we add the comma after the last digit of the number whose root we are looking for. Moreover, after the comma, we add a number of pairs of zeros equal to the number of decimal places we want to compute for the root.
Par example, suppose we want to calculate the square root of 179, with two decimals. In this case, the comma being placed after the last digit, two pairs of zeros are added. Then we continue the calculations in the usual way.
The result 13,37 is the approximate value, calculated to the second decimal place, of the square root of 179. To verify this calculation, proceed as in the previous case, that is to say that the we multiply 13.37 x 13.37.
The result of this operation is 178.7569 ; as we see, it is not equal to 179, since the number 13,37 is only the approximate value of the root. To complete the verification, we must add the value of the rest :
178,7569 + 0,2431 = 179
The result of this last operation, being equal to the number given at the beginning of the calculations, confirms the accuracy of all the operations.
Another example with another provision : Calculate the square root of 0.00027, we write :
As this example shows, the integer part of the number is zero, so the integer part of the root is 0. The first group to the right of the decimal point is two zeros, so the first decimal digit of the root is zero. The second group to the right of the comma is 02, which is 2. The largest square contained in 2 is 1 ; its root is 1, and this one will be the second decimal digit of the searched root.
Root extraction continues normally. As we have seen, extracting the root is a somewhat tedious operation.
On the other hand, even squaring, which is very simple, can make a calculation made up of many other operations tedious ; commercially available machines are now available at a reasonable or derisory price, with scientific calculating machines making it possible to immediately find the square root or the square of a number.
Square roots
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