Example 1 Compare 6*√7 and 7*√6. Analyze and move the factors outside the root symbol into the root symbol, and then
Compare the size of the roots. The answer is because 6*6*7=252, 7*7*6, and 252.
6 *√7 & lt； 7*√6.
Two-stage method
Example 2( 1) Try to compare the sizes of √5+√ 13, √7+√ 1 1 and √8+ 10;
(Analysis and observation show that each group is the sum of two quadratic roots, which can be squared before comparison;
(2) Further observation shows that the sum of two quadratic roots in each group is equal, and the two roots are getting closer and closer. Considering the relationship between the size of each group, we can make a guess.
Brief introduction of solution
Guess: If 0
3 comparison method
Compare √ 2003-1√ 2004-1and √ 2003+1√ 2004+1.
It is observed that the denominator of the two formulas can be multiplied by the square difference formula, and the result is an integer, so the difference is made.
Contrast. (Do it yourself, typing it out is really troublesome.)
4 than commercial law
Example 4 compares the offsets of √ A+1√ A+4 and √a+2/√a+3.
Analysis and observation, this problem can still be used as "relatively poor".
Method "is relatively large, but for comparison, the calculation is also very square.
Very convenient.
Brief introduction of solution
5. Physical and chemical methods
This should be familiar, so I won't explain it much.
6-to-middle method
Example 7 0
The analysis is determined by the condition 0 1, √ 1+( 1-x) 2 > L, so √1+x 2+√1-x) 2 > 2,
7 special value method
The simplest and most practical method
Figure-eight combination method
Not commonly used, no examples, if necessary.
9 Using known inequalities
Several basic inequalities in high school textbooks
10 uses scaling transformation.
Implicit condition of Ll application
Compare the cube of 8-m with the size of √m- 15.
The analysis of this problem has the implicit condition m- 15≥0, so m≥ 15, from 8-m.