Instruction

1

Decompose any two-digit

**number**into components, highlighting the number of units. Among the 96 number of units — 6. Therefore, we can write 96 = 90 + 6.2

Erect in

**the square**the first number: 90 * 90 = 8100.3

Do likewise with the second

**number**m: 6 * 6 = 364

Multiply the number between them and double the result: 90 * 6 * 2 = 540 * 2 = 1080.

5

Add the results of the second, third and fourth steps: 8100 + 36 + 1080 = 9216. This is the result of raising in

**the square**of the number 96. After some training you will be able to quickly take steps in the mind, surprising parents and classmates. Until I mastered it, record the results of each step to avoid confusion.6

For training lift in

**the square****the number is**74 check yourself on the calculator. The sequence of actions: 74 = 70 + 4, 70 * 70 = 4900, 4 * 4 = 16, 70 * 4 * 2 = 560, 4900 + 16 + 560 = 5476.7

Raise to the second power

**of the number**81. Your actions: 81 = 80 + 1, 80 * 80 = 6400, 1 * 1 = 1, 80 * 1 * 2 = 160, 6400 + 1 + 160 = 6561.8

Remember special a method of construction in

**square**two digit numbers ending in the digit 5. Highlight the number of tens in 75 of their 7 pieces.9

Multiply the number of tens to the next figure of

**the number**of CMV series: 7 * 8 = 56.10

Assign the right

**number**25: 5625 — the result of raising in**the square**of the number 75.11

For training lift in the second degree

**the number**95. It ends in the digit 5, so the sequence of actions: 9 * 10 = 90, 9025 — result.12

Learn how to build in

**the square**of a negative number: -95 in**the square**e = 9025, as in the eleventh step. Similarly -74 in**the square**e = 5476, as in the sixth step. This is because when multiplying two negative numbers always get a positive**number**: -95 * -95 = 9025. Therefore, in the construction in the**square**can just not pay attention to the minus sign.Useful advice

To exercise was not boring, call on the help of a friend. He writes two-digit number, and you — the result of the construction of the number in the square. Then switch places.

# Advice 2 : As you say " to the degree

The construction of the degree is one of the simplest algebraic operations. In everyday life the construction is rare, but on production, when you perform calculations – virtually everywhere, so it is useful to recall how this is done.

Instruction

1

Suppose we have some number a, the degree of which is the number n. To build a number to a power means to multiply the number a to itself n times.

2

Consider a few examples.

To build the number 2 in the second degree, you must produce action:

2x2=4

To build the number 2 in the second degree, you must produce action:

2x2=4

3

To build the number 3 in the fifth degree, you must take action:

3х3х3х3х3=243

3х3х3х3х3=243

4

There is no generally accepted designation of the second and third degree numbers. The phrase "second degree" is usually replaced by the word "square" instead of the phrase "the third degree" usually say "cube".

5

As can be seen from the above examples, the duration and the complexity of calculations depends on the magnitude of the exponent of the number. Square or cube – a relatively easy task; raising a number to the fifth or greater degree already requires more time and accuracy in calculations. To accelerate this process and to avoid errors, you can use a special mathematical tables or a scientific calculator.

# Advice 3 : How to put in 1 degree

For a brief record of the works of one and the same number by itself, mathematicians have invented the concept of degree. Therefore, the expression 16*16*16*16*16 you can write the shorter way. It would be 16^5. The expression will be read as number 16 in the fifth degree.

You will need

- Paper, pen.

Instruction

1

In General, the

The expression a^n is called

a is the number of base degree,

n is a number, the exponent. For example, a = 4, n = 5,

Then write 4^5 = 4*4*4*4*4 = 1 024

**degree**is written as a^n. This entry means that the number a multiplied by itself n times.The expression a^n is called

**the degree of**u,a is the number of base degree,

n is a number, the exponent. For example, a = 4, n = 5,

Then write 4^5 = 4*4*4*4*4 = 1 024

2

The degree n can be a negative number

n = -1, -2, -3, etc.

To compute the negative

a^(-n) = (1/a)^n = 1/a*1/a*1/a* ... *1/a = 1/(a^n)

Consider the example

2^(-3) = (1/2)^3 = 1/2*1/2*1/2 = 1/(2^3) = 1/8 = 0,125

n = -1, -2, -3, etc.

To compute the negative

**degree**numbers must be omitted in the denominator.a^(-n) = (1/a)^n = 1/a*1/a*1/a* ... *1/a = 1/(a^n)

Consider the example

2^(-3) = (1/2)^3 = 1/2*1/2*1/2 = 1/(2^3) = 1/8 = 0,125

3

As can be seen from the example, -3

1) First calculate the fraction 1/2 = 0,5; and then to build in

ie 0,5^3 = 0,5*0,5*0,5 = 0,125

2) First build in the denominator

**the degree**of the number 2 can be calculated in different ways.1) First calculate the fraction 1/2 = 0,5; and then to build in

**a degree of**3,ie 0,5^3 = 0,5*0,5*0,5 = 0,125

2) First build in the denominator

**degree**2^3 = 2*2*2 = 8, and then calculate the fraction 1/8 = 0,125.4

Now compute -1

a^(-1) = (1/a)^1 = 1/(a^1) = 1/a

For example, let's build the number 5 to -1

5^(-1) = (1/5)^1 = 1/(5^1) = 1/5 = 0,2.

**degree**for the number, i.e., n = -1. The rules discussed above are suitable for this case.a^(-1) = (1/a)^1 = 1/(a^1) = 1/a

For example, let's build the number 5 to -1

**degree**5^(-1) = (1/5)^1 = 1/(5^1) = 1/5 = 0,2.

5

From the example clearly shows that the number -1 is the inverse fraction of the number.

Imagine the number 5 as a fraction 5/1, then 5^(-1) arithmetically not take it and immediately write the inverse of 5/1 is 1/5.So, 15^(-1) = 1/15,

6^(-1) = 1/6,

25^(-1) = 1/25

Imagine the number 5 as a fraction 5/1, then 5^(-1) arithmetically not take it and immediately write the inverse of 5/1 is 1/5.So, 15^(-1) = 1/15,

6^(-1) = 1/6,

25^(-1) = 1/25

Note

When raising a number to a negative exponent, it should be remembered that the number can't be zero. According to the rule, we need to lower the number in the denominator. And zero cannot be the denominator, because zero cannot be split.

Useful advice

Sometimes when working with degrees for ease of calculation, the fractional number of specially replace the integer to -1 degree

1/6 = 6^(-1)

1/52 = 52^(-1).

1/6 = 6^(-1)

1/52 = 52^(-1).

# Advice 4 : How to put a fraction into a square

In the solution of arithmetic and algebraic tasks is sometimes required to build

**a fraction**in**a square**. The easiest way to do it when**the fraction**to a decimal is fairly simple calculator. However, if**the fraction**of the ordinary or combined, in the construction of such number in**the square**may experience some difficulties.You will need

- calculator, computer, Excel.

Instruction

1

To build a decimal

**fraction**in**the square**, take a scientific calculator, type it erected in**the square of****the fraction**, and press the construction in the second degree. On most calculators this button is labeled as "x2". On the standard Windows calculator function calculates the**square**looks like "x^2". For example,**square**a decimal, is equal to 3,14: 3,142 = 9,8596.2

To build in

**the square**of a decimal**fraction**on a normal (accounting) a calculator, multiply this number with itself. By the way, some models of calculators the possibility of raising the number in**the square**even in the absence of a special button. So please check the instruction manual for the specific calculator. Sometimes examples of "tricky" exponentiation is given on the back cover or on the box of the calculator. For example, many calculators for the erection of a number in**a square**is enough to press button "x" and "=".3

For raising to

**the square**fractions (consisting of numerator and denominator), erected in**the square**separately the numerator and denominator of this fraction. That is, use the following rule:(b / h)2 = P2 / Z2, where h is the numerator, b – denominator.Example: (3/4)2 = 32/42 = 9/16.4

If erected in

**the square of****the fraction**– combined (composed of a whole part and fractions), you must bring her to the ordinary mind. That is, apply the following formula:(C h/z)2 = ((u*h+h) / h)2 = (p*z+h)2 / Z2, where C is the integer part of mixed fraction.Example: (3 2/5)2 = ((3*5+2) / 5)2 = (3*5+2)2 / 52 = 172 / 52 = 289/25 = 11 14/25.5

If you build in

**a square**ordinary (not decimal) fractions you have to constantly, use the program MS Excel. To do this, enter in one cell the following formula: =DEGREE(A2;2) where A2 is the cell address which will be entered erected in**the square****a fraction**.To tell the program that input numbers must be treated as an ordinary**fraction**(i.e., not to convert it to decimal form), dial before you**roll**the first digit "0" and "gap". That is, for input, for example, the fraction 2/3, you need to enter: "0 2/3" (and press Enter). While in the entry line displays the decimal representation of fractions introduced. The meaning and representation of fractions directly in the cell preserved in its original form. In addition, when using mathematical functions, whose arguments are fractions the result will also be presented in the form of fractions. Hence**the square**of the fraction 2/3 would be represented as 4/9.# Advice 5 : How to be a square of binomials

The method of allocation of square binomials used in simplifying the cumbersome expressions, and solving quadratic equations. In practice it is usually combined with other techniques, including factorization, grouping, etc.

Instruction

1

The method of singling out a complete square of binomials is based on the use of two formulas of the reduced multiplication of polynomials. These formulas are special cases of the Binomial theorem for the second degree and allow you to simplify the search expression so that it was possible to conduct further reduction or decomposition on the multipliers:

(m + n)2 = m2 + 2·m·n + n2;

(m - n)2 = m2 - 2·m·n + n2.

(m + n)2 = m2 + 2·m·n + n2;

(m - n)2 = m2 - 2·m·n + n2.

2

According to this method from the original polynomial is required to allocate the squares of the two monomials and the sum/difference of their double works. The application of this method makes sense if the high-degree terms not less than 2. Suppose the task is to factorize a lowering of the degree the following expression:

4·y^4 + z^4

4·y^4 + z^4

3

To solve the problem you need to use the method of allocation of the full square. Thus, the term consists of two variables with monomials of even degree. Therefore, it is possible to identify each of them using m and n:

m = 2·y2; n = z2.

m = 2·y2; n = z2.

4

Now we need to give initial expression to the form (m + n)2. It already contains the squares of these terms, but not enough double work. You need to add it artificially, and then subtract:

(2·y2)2 + 2·2·y2·z2 + (z2)2 - 2·2·y2 ·z2 = (2·y2 + z2)2 – 4·y2·z2.

(2·y2)2 + 2·2·y2·z2 + (z2)2 - 2·2·y2 ·z2 = (2·y2 + z2)2 – 4·y2·z2.

5

In the resulting expression, you can see the formula difference of squares:

(2·y2 + z2)2 – (2·y·z)2 = (2·y2 + z2 – 2·y·z)· (2·y2 + z2 + 2·y·z).

(2·y2 + z2)2 – (2·y·z)2 = (2·y2 + z2 – 2·y·z)· (2·y2 + z2 + 2·y·z).

6

So, the method consists of two stages: allocation of the monomials of a complete square of m and n, the addition and subtraction of their double works. Selection method full square binomials can be used not only independently but also in combination with other methods: making the brackets common factor, change of variable, grouping terms, etc.

7

Example 2.

Select the full square in the expression:

4·y2 + 2·y·z + z2.

Solution.

4·y2 + 2·y·z + z2 =[m = 2·y, n = z] = (2·y)2 + 2·2·y·z + (z) 2 – 2·y·z = (2·y + z)2 – 2·y·z.

Select the full square in the expression:

4·y2 + 2·y·z + z2.

Solution.

4·y2 + 2·y·z + z2 =[m = 2·y, n = z] = (2·y)2 + 2·2·y·z + (z) 2 – 2·y·z = (2·y + z)2 – 2·y·z.

8

The method used for finding roots of a quadratic equation. The left part of the equation is a trinomial of the form a·y2 + b·y + c, where a, b and c are numbers and a ≠ 0.

a·y2 + b·y + c = a·(y2 + (b/a)·y) + c = a·(y2 + 2·(b/(2·a))·y) + c = a·(y2 + 2·(b/(2·a))·y + b2/(4·a2)) + c – b2/(4·a) = a·(y + b/(2·a)) 2 – (b2 – 4·a·c)/(4·a).

a·y2 + b·y + c = a·(y2 + (b/a)·y) + c = a·(y2 + 2·(b/(2·a))·y) + c = a·(y2 + 2·(b/(2·a))·y + b2/(4·a2)) + c – b2/(4·a) = a·(y + b/(2·a)) 2 – (b2 – 4·a·c)/(4·a).

9

These calculations lead to the notion of discriminant which is equal to (b2 – 4·a·c)/(4·a) and the roots of the equation are equal:

y_1,2 = ±(b/(2•a)) ± √ ((b2 – 4·a·c)/(4·a)).

y_1,2 = ±(b/(2•a)) ± √ ((b2 – 4·a·c)/(4·a)).

# Advice 6 : How to put a negative number in degree

The operation of raising to

**a degree**is binary, i.e. has two required input arguments and one output. One of the initial parameters is called the exponent determines the number of times that a multiplication operation needs to be applied to the second parameter - the base. The base can be both positive and negative**numbers**.Instruction

1

Use the exponentiation of a negative number are common to the operation of the rule. As for positive integers, exponentiation means multiplying the original value by itself the number of times per unit less the exponent. For example, to build in the fourth degree, the number -2, it must three times to multiply by itself: -2⁴=-2*(-2)*(-2)*(-2)=16.

2

Multiplying two negative numbers always gives a positive value, and the result for values with different signs is negative. From this we can conclude that in the construction of negative values in the degree with even index should always be positive, and at odd indices of the result will always be less than zero. Use this property to check of the calculations. For example, the -2 in the fifth degree should be a negative number -2⁵=-2*(-2)*(-2)*(-2)*(-2)=-32, and -2 in the sixth, positive -2⁶=-2*(-2)*(-2)*(-2)*(-2)*(-2)=64.

3

During the construction of the negative of the power index can be given in the format of fractions, for example, -64 to degree⅔. This figure means that the original value be raised to the degree equal to the numerator, and extract the root degree equal to the denominator. One part of this operation, considered in the previous steps, but here you should pay attention to another.

4

The root is an odd function, that is, for negative real numbers it can only be used for odd exponent. Even if this feature does not matter. So, if you need to build a negative number to a fractional degree with an even denominator, then the problem has no solution. In all other cases, do the first transaction from the first two steps, using as the exponent of the numerator of the fraction, and then extract the root with the degree of the denominator.