(1) When a, b, c and d satisfy any conditions, s is a rational number;
(2) When A, B, C and D satisfy any condition, S is an irrational number.
①5√8-2√32+√50
=5*3√2-2*4√2+5√2
=√2( 15-8+5)
= 12√2
②√6-√3/2-√2/3
=√6-√6/2-√6/3
=√6/6
③(√45+√27)-(√4/3+√ 125)
=(3√5+3√3)-(2√3/3+5√5)
=-2√5+7√5/3
④(√4a-√50b)-2(√b/2+√9a)
=(2√a-5√2b)-2(√2b/2+3√a)
=-4√a-6√2b
⑤√4x*(√3x/2-√x/6)
=2√x(√6x/2-√6x/6)
=2√x*(√6x/3)
=2/3*|x|*√6
⑥(x√y-y√x)÷√xy
=x√y÷√xy-y√x÷√xy
=√x-√y
⑦(3√7+2√3)(2√3-3√7)
=(2√3)^2-(3√7)^2
= 12-63
=-5 1
⑧(√32-3√3)(4√2+√27)
=(4√2-3√3)(4√2+3√3)
=(4√2)^2-(3√3)^2
=32-27
=5
⑨(3√6-√4)? 0? five
=(3√6)^2-2*3√6*√4+(√4)^2
=54- 12√6+4
=58- 12√6
⑩( 1+√2-√3)( 1-√2+√3)
=[ 1+(√2-√3)][ 1-(√2-√3)]
= 1-(√2-√3)^2
= 1-(2+3+2√6)
=-4-2√6
1 1.√( 1/2x)^2+ 10/9x^2
√[( 1/2x)^2+ 10/9x^2]
=√(x^2/4+ 10x^2/9)
=√(9x^2/36+40x^2/36)
=√(49x^2/36)
= 7 | x |/6；
12.4mb 2n+ 1 (A and B are positive numbers)
[√ (A 4mb 2n)]+ 1 (A and B are both positive numbers)
=a^2mb^n+ 1；
13.
√(4a^5+8a^4)(a^2+3a+2)(a>； =0)
√[(4a^5+8a^4)(a^2+3a+2)](a>； =0)
=√[4a^4(a+2)(a+2)(a+ 1)]
=√[(2a^2)^2(a+2)^2(a+ 1)]
=2a^2(a+2)√(a+ 1).
14. When-1
Root number (x-3)2+ root number (x+ 1) 2
=|x-3|+|x+ 1|
=3-x+x+ 1
=4
15.
simplify
Root number (a2b)+ root number (a-b)2+ root number (c2b)+ root number (b^2ac)
=|A| Root number B+|A-B|+|C| Root number B+|B| Root number AC
The following questions: questions 16 to 19.
(1) proves that the square root is meaningful no matter what value m takes.
(2) When finding the value of m, the quadratic formula √ 27-4m+2m 2 has a minimum value.
16.
If √ x-∏+∏-x+ (absolute value 2y- 1)=5, then x= y=
Because the root sign is greater than or equal to 0.
So x-π >; =0，π-x & gt； =0
X-π and π-x are opposites.
Are all greater than or equal to 0, so they can only be equal to 0.
So x-π=0, x=π.
So 0+0+|2y- 1|=5.
2y- 1=5 or 2y- 1=-5, y=3, y=-2.
So x=π, y=-2 or x=π, y=3.
17.
If √x+y(√(x+y)- 1)=2, find the value of x+y.
Make √(x+y)=a
Then a(a- 1)=2.
a^2-a-2=0
(a-2)(a+ 1)=0
a=- 1，a=2
Because the root sign is greater than or equal to 0.
So a=- 1 is discarded.
So √(x+y)=a=2.
18.
Given x > 0.y > 0, x+3√xy-4y=0, find the value of √ x: √ y.
x+3√xy-4y=0
(√x)^2-3√x*√y-4(√y)^2=0
(√x-4√y)(√x+√y)=0
Because x>0, y>0
So √ x > 0，√y & gt； 0
√x+√y & gt； 0
So √x+√y=0 is not valid.
So √x-4√y=0.
√x=4√y
√x:√y=4
19.
Known quadratic formula √ 27-4m+2m 2
2m^2-4m+27
=2(m^2-2m+ 1)+25
=2(m- 1)^2+25
Because (m- 1) 2 > =0.
So 2 (m- 1) 2 >: =0
2(m- 1)^2+25>； = 25 & gt0
So no matter what value m takes, 2m 2-4m+27 is greater than 0.
So the square root makes sense.
The minimum value of (m- 1) 2 is 0, and m= 1.
So when m= 1, the square root √ 27-4m+2m 2 has a minimum value.
In addition:
20.
If the tenth power of x under the radical sign is the simplest quadratic radical (x is not equal to 0), then mn=?
√[(x/m)^n]
A quadratic radical satisfying the following conditions is called the simplest quadratic radical:
The factor of (1) root sign is an integer and the factor is an algebraic expression;
(2) The number of square roots does not contain factors or factors that can be opened to the maximum.
√[(x/m)^n]
Is the simplest quadratic root:
Obviously, n= 1 (excluding factor or the factor that can be opened to the maximum, so it must be less than 2).
M= 1 (the factor of the root sign is an integer and the factor is an algebraic expression).
Mn= 1 There is an answer! !