How do you calculate a quadratic radical and how do you calculate a quadratic radical?

Quadratic radical. i.Definitions:

A formula of the form ā(a 0) is called a quadratic radical.

ii.The range of quadratic radical ā.

is a non-negative number. i.e. ā 0.

When a 0, ā denotes the arithmetic square root of a.

When a=0, ā represents the arithmetic square root of 0, i.e., 0.

iii.Calculation formula:

1.（√=a（a≥0）

2.When a 0, ā=a

When a=0, ā=0

When a 0, ā=a

3. √=√āōa≥0, o≥0）

√=√(āa≥0, o≥0）

iv.The simplest quadratic radical.

Conditions: (1) The number of squares to be opened does not contain a denominator; (2) The number of squares to be opened does not contain a factor that can be opened to the end.

v.Addition and subtraction of quadratic radicals.

First, the quadratic radicals are reduced to the simplest quadratic radicals, and then the radicals with the same number of squares are merged.

Note: Quadratic radicals have double non-negative properties

Multiplication of quadratic radicals:

1) Rule: Root A · Root B = Root Ab (A 0 and B 0).

2) Type: single quadratic radical multiplied by single quadratic radical;

A single quadratic radical multiplied by a multiple-quadratic radical;

Multinomial quadratic radicals multiplied by multinomial quadratic radicals.

When performing multiplication operations, multiplication formulas can sometimes be applied to make the calculation easy.

3.Quadratic radical division:

1) Rule: Root A · Root B = Root Ab (A 0 and B 0).

2) Type: Single quadratic radical divided by single quadratic radical (calculated by applying algorithm).

The multinomial quadratic radical divided by the single quadratic radical ** is converted into the single quadratic radical divided by the single quadratic radical).

The divisor is the sum of two quadratic radicals or the sum of a quadratic radical and a rational number (the denominator is rationalized and calculated, or the operation of the fraction is thought of by analogy, and the numerator and the common factor in the denominator are removed).

The quadratic root is the inverse of the square of the open square.

1. Addition and subtraction of quadratic radicals:

First, the quadratic radicals in the formula are reduced to the simplest quadratic radicals, and then the parentheses are removed and the similar quadratic radicals are merged with the addition and subtraction of polynomials.

2. Multiplication of quadratic radicals:

1) Rule: Root A·Root B = Root Ab (A 0 and B 0) (2) Type:

i) a single quadratic radical multiplied by a single quadratic radical;

ii) a single quadratic radical multiplied by a multiple-second quadratic radical;

iii) multiplying multiple quadratic radicals by multiple quadratic radicals.

1. Determine the order of operations.

2. Flexible use of the law of operation.

3. Use multiplication formulas correctly.

4. Most of the denominators should be rationalized in a timely manner.

5. In some simple calculations, it may be possible to reduce the fraction, do not blindly rationalize (but the final result must be the denominator of the rational family).

6. Pay attention to the implicit conditions and the indication of the parentheses at the end when operating letters.

7. When mentioning the common factor, you can consider mentioning the common factor closure with the root number.

1 Every two digits from the unit digit to the left, if there is a decimal place every two quarters from the decimal point to the right, use the "," sign to separate the sections;

2. Find the perfect square number that is not greater than the number of the first section on the left, which is "quotient";

3 Subtract the quotient obtained from the first section on the left, and write the second section as the first remainder to the right of their difference.

4. Multiply the quotient by 20 and divide the first remainder to obtain the largest integer as the quotient (if the maximum integer is greater than or equal to 10, use 9 or 8 as the quotient);

5 Multiply the quotient by 20 plus the test quotient and multiply by the test quotient. If the product obtained is less than or equal to the remainder, write this trial quotient after the quotient as a new quotient; If the product obtained is greater than the remainder, the test quotient will be reduced one by one until the product is less than or equal to the remainder;

6 In the same way, keep asking.

The above method of writing and prescribing is the method given in the appendix of the textbook when most of us go to school, and in practice it is too troublesome to calculate. We can take the following methods, and we are not afraid of a mistake in the actual calculation!! The above method does not work.

For example, 136161 this number, first we find a number that is close to the square root of the 136161, and choose any one, for example, any number between 300 and 400, here we choose 350 as a representative.

We calculated.

Then we do the math and we find that the sum is almost the same, and that 369 2 has the last digit of 1. We have reason to conclude that 369 2 = 136161

Generally speaking, if you can open as much as possible, the basic results will come out after one or two calculations using the above methods. Another example: calculate the square root of a 469225.

First we find 600 2 <469225< 700 2, and we can pick 650 as the number to calculate for the first time. Count.

Get. And around 685, only 685 2 has a number at the end of 5, so 685 2 = 469225

For those numbers that are inexhaustible, the accuracy of two or three times in this method is considerable, generally reaching several decimal places.

In practice, this algorithm is also the algorithm used by computers to prescribe.

Resources.

Multiplication of quadratic radicals:

1) Rule: Root A · Root B = Root Ab (A 0 and B 0) 2) Type: Single quadratic radical multiplied by single quadratic radical;

A single quadratic radical multiplied by a multiple-quadratic radical;

Multinomial quadratic radicals multiplied by multinomial quadratic radicals.

When performing multiplication operations, it is sometimes possible to apply the multiplication Youwu Li formula to make the calculation simple.

3.Quadratic radical division:

1) Rule: Root A Root B = Root A B (A 0 and B >0) 2) Type: Single quadratic radical divided by single quadratic radical (calculated by applying the algorithm) Multiple quadratic radicals divided by single quadratic radical ** into single quadratic radical divided by single quadratic radical) The divisor is the sum of two quadratic radicals or the sum of a quadratic radical and a rational number (the denominator is rationalized and calculated, or the operation of the fraction is analogous to think, reduce the numerator, the common factor in the denominator)

√12-√27-√20+√50

2 roots, 3-3 roots, 3-2 roots, 5+5 roots, 2

Root No. 3-2, Root No. 5+5, Root No. 2

4=2！The square root of the key to 4 is 2, and the square of 2 is 4! 4= 2, which means that the square root of 4 is 2, i.e. +2 or -2 is 4!