16-Ra\303\255zes Quadradas Examinaremos as no\303\247\303\265es da raiz quadrada gr\303\241fica e numericamente, para ent\303\243o usar essa compreens\303\243o para simplificar os n\303\272meros e express\303\265es em raiz quadrada. Estudaremos tamb\303\251m expoentes fracion\303\241rios de ra\303\255zes quadradas.

restart; with(plots): with(RealDomain):

Warning, the name changecoords has been redefined Warning, these protected names have been redefined and unprotected: Im, Re, ^, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, eval, exp, expand, limit, ln, log, sec, sech, signum, simplify, sin, sinh, solve, sqrt, surd, tan, tanh

Gr\303\241ficoPadr\303\243o := proc( number ) local N, n, A, h, v, i, ps; N := floor(number): n := ceil( sqrt(N)): A := plot( sqrt(x), x = -1..(N+1), tickmarks = [N-1,n-1] ): h := seq(plot( [[0,k],[N,k]], color = blue), k = 1..n): v := seq(plot( [[k,0],[k,n]], color = blue), k = 1..N): for i from 0 to N do if( floor( sqrt(i)) = sqrt(i) ) then ps||i := plottools[disk]([i, sqrt(i)], .25, color=red): else ps||i := plottools[disk]([i, sqrt(i)], .15, color=yellow): fi: od: plots[display]( { A,h,v,seq(ps||i, i = 0..N)}, scaling=constrained ); end proc:

Se olharmos o gr\303\241fico de uma raiz quadrada em uma grade, veremos que a curva passa apenas por alguns pontos da grade. StandardPlot(10);

NiMtJS1TdGFuZGFyZFBsb3RHNiMiIzU=

em particular, a curva passa pelas interse\303\247\303\265es em x = 0, 1, 4, 9. Vejamos um diagrama grande desse tipo. StandardPlot(27);

NiMtJS1TdGFuZGFyZFBsb3RHNiMiI0Y=

Vemos os pontos vermelhos em x = 0,1, 4, 9, 16, 25. Parece que a raiz quadrada tem valores inteiros apenas para valores que s\303\243o quadrados perfeitos.

Esta \303\251 uma tabela de quadrados perfeitos. A coluna da esquerda \303\251 um n\303\272mero e a coluna da direita o seu quadrado.Voce deve reconhecer os n\303\272meros na coluna da direita como os mesmos que a curva da raiz quadrada cruzou as interse\303\247\303\265es da grade. array( [seq( [ k, k^2 ],k = 1..18)]);

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

Qual o n\303\272mero, que elevado ao quadrado d\303\241 121? A resposta a essa pergunta \303\251 a "Raiz quadrada" - o inverso do quadrado. sqrt(121);

NiMiIzY=

E, logicamente, se elevarmos ao quadrado, voltamos ao n\303\272mero original. 11^2;

NiMiJEAi

Que n\303\272mero, quando elevado ao quadrado, d\303\241 324? sqrt(324);

NiMiIz0=

Que n\303\272mero, quando elevado ao quadrado, d\303\241 2500? sqrt(2500);

NiMiI10=

Se escrevermos esses n\303\272meros na dire\303\247\303\243o oposta, teremos uma tabela de raiz quadrada. Isso mostra que a raiz quadrada de um n\303\272mero que \303\251 quadrado \303\251 o pr\303\263prio n\303\272mero original.. array( [seq( [ k^2 , k],k = 1..13)]);

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

E quanto \303\240s ra\303\255zes quadradas de outros n\303\272meros, que n\303\243o s\303\243o quadrados perfeitos? A raiz quadrada de um n\303\272mero inteiro \303\251 outro n\303\272mero inteiro ou um n\303\272mero irracional. array( [seq( [ k , sqrt(k) = evalf( sqrt(k), 15) ],k = 1..13) ]);

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

Por isso, \303\251 uma boa ideia memorizar alguns quadrados perfeitos, pois pode ser \303\272til no futuro. array( [seq( [ k , k^2],k = 1..20) ]);

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array( [seq( [ 5*k , (5*k)^2],k = 1..20) ]);

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E quando o n\303\272mero n\303\243o \303\251 um quadrado perfeito, voc\303\252 pode deix\303\241-lo na forma de radical, ou usar o Maple para achar parte do n\303\272mero decimal infinito. sqrt( 101): % = evalf(%,20);

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sqrt( 214): % = evalf(%,20);

NiMvKiQpIiQ5IyMiIiIiIiMiIiIkIjVkTXpGJFEpUShHWSIhIz0=

sqrt( 1333): % = evalf(%,20);

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sqrt( 123349232401 ): % = evalf(%,20);

NiMvKiQpIi0sQ0JcTDcjIiIiIiIjIiIiJCI1Q3cwdyp5JDQ2N04hIzk=

Quando um n\303\272mero \303\251 quadrado perfeito, a sua raiz quadrada \303\251 simplesmente o n\303\272mero original. sqrt(16); sqrt(51^2);

NiMiIiU=NiMiI14=

Este mesmo princ\303\255pio \303\251 aplicado \303\240s vari\303\241veis. A raiz quadrada de um quadrado perfeito \303\251 o valor absoluto no n\303\272mero original.. sqrt( x^2): % = simplify( %);

NiMvKiQpKiQpJSJ4RyIiIyIiIiMiIiIiIiMiIiItJSRhYnNHNiMlInhH

quando um n\303\272mero nao \303\251 um quadrado perfeito, n\303\243o h\303\241 nada a fazer, pois ele n\303\243o possui raiz quadrada exata. sqrt( 3*5*11);

NiMqJCkiJGwiIyIiIiIiIyIiIg==

Ou podem ser quadrados perfeitos disfar\303\247ados - que n\303\243o s\303\243o sempre \303\263bvios. sqrt( 3^3 ); sqrt( 27 );

NiMsJComIiIkIiIiKSIiJCMiIiIiIiMiIiIiIiI=NiMsJComIiIkIiIiKSIiJCMiIiIiIiMiIiIiIiI=

sqrt( 2*25); sqrt( 50 );

NiMsJComIiImIiIiKSIiIyMiIiIiIiMiIiIiIiI=NiMsJComIiImIiIiKSIiIyMiIiIiIiMiIiIiIiI=

sqrt( (2^5)*(3^4)*(5^3)*7);

NiMsJComIiQhPSIiIikiI3EjIiIiIiIjIiIiIiIi

Perceba que o Maple representa uma raiz quadrada como um n\303\272mero elevado \303\240 meia pot\303\252ncia. 3^(1/2);

NiMqJCkiIiQjIiIiIiIjIiIi

Isso porque \303\251 a mesma coisa. Perceba que se elevarmos tanto a raiz quadrada quanto o expoente fracion\303\241rio, teremos o mesmo resultado. sqrt( a )^2; (a^(1/2))^2;

NiMlImFHNiMlImFH