PROVE THAT (1+A2)(1+B2)≥4AB FOR ALL A,B>0 WITHOUT LIMITS

     

This solution giao dịch with adding, subtracting & finding the least common multiple.




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Step by Step Solution

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Step 1 :

1 Simplify — aEquation at the over of step 1 : 1 1 1 ————-———— ÷ (—•b) (a2) (b2) a

Step 2 :

1 Simplify —— b2Equation at the over of step 2 : 1 1 b ———— - —— ÷ — (a2) b2 a

Step 3 :

1 b Divide —— by — b2 a3.1 Dividing fractions khổng lồ divide fractions, write the divison as multiplication by the reciprocal of the divisor :

1 b 1 a—— ÷ — = —— • —b2 a b2 bMultiplying exponential expressions :3.2 b2 multiplied by b1 = b(2 + 1) = b3

Equation at the end of step 3 :

1 a ———— - —— (a2) b3

Step 4 :

1 Simplify —— a2Equation at the end of step 4 : 1 a —— - —— a2 b3

Step 5 :

Calculating the Least Common Multiple :5.1 Find the Least Common Multiple The left denominator is : a2 The right denominator is : b3

Number of times each Algebraic Factorappears in the factorization of:AlgebraicFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right
a202
b033

Least Common Multiple: a2b3

Calculating Multipliers :

5.2 Calculate multipliers for the two fractions Denote the Least Common Multiple by L.C.M Denote the Left Multiplier by Left_M Denote the Right Multiplier by Right_M Denote the Left Deniminator by L_Deno Denote the Right Multiplier by R_DenoLeft_M=L.C.M/L_Deno=b3Right_M=L.C.M/R_Deno=a2

Making Equivalent Fractions :

5.3 Rewrite the two fractions into equivalent fractionsTwo fractions are called equivalent if they have the same numeric value. For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well. Lớn calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L.

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Num. B3 —————————————————— = ———— L.C.M a2b3 R. Mult. • R. Num. A • a2 —————————————————— = —————— L.C.M a2b3 Adding fractions that have a common denominator :5.4 Adding up the two equivalent fractions add the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce lớn lowest terms if possible:

b3 - (a • a2) b3 - a3 ————————————— = ——————— a2b3 a2b3 Trying khổng lồ factor as a Difference of Cubes:5.5 Factoring: b3 - a3 Theory : A difference of two perfect cubes, a3-b3 can be factored into(a-b)•(a2+ab+b2)Proof:(a-b)•(a2+ab+b2)=a3+a2b+ab2-ba2-b2a-b3 =a3+(a2b-ba2)+(ab2-b2a)-b3=a3+0+0-b3=a3-b3Check: b3 is the cube of b1Check: a3 is the cube of a1Factorization is :(b - a)•(b2 + ab + a2)

Trying to factor a multi variable polynomial :5.6 Factoringb2 + ab + a2Try to factor this multi-variable trinomial using trial and errorFactorization fails