8-Regras Exponenciais Estudaremos aqui as regras dos expoentes e como resolver problemas envolvendo expoentes.

restart; txln := `\134n____________________________________________________________________________`:

Veremos algumas das principais regras de expoentes e as suas devidas aplica\303\247\303\265es no Maple: print(cat(txln,`\134n1. Regra de Produto`)); a^n * a^m: %= simplify(%);print(txln);

NiMlXHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzEuflJlZ3JhfmRlflByb2R1dG9HNiMvKiYpJSJhRyUibkciIiIpJSJhRyUibUciIiIpJSJhRywmJSJuRyIiIiUibUciIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

10^5 * 10^2;

NiMiKSsrKzU=

x^5 * x^4;

NiMqJCklInhHIiIqIiIi

y^8 * y^5;

NiMqJCklInlHIiM4IiIi

(x+4)^10 * (x+4)^8;

NiMqJCksJiUieEciIiIiIiUiIiIiIz0iIiI=

x^4 * y^3; `Note que as bases podem ser as mesmas que usadas nessa regra. `;

NiMqJiklInhHIiIlIiIiKSUieUciIiQiIiI=NiMlaW5Ob3RlfnF1ZX5hc35iYXNlc35wb2RlbX5zZXJ+YXN+bWVzbWFzfnF1ZX51c2FkYXN+bmVzc2F+cmVncmEufkc=

distance^2 * distance^3;

NiMqJCklKWRpc3RhbmNlRyIiJiIiIg==

print(cat(txln,`\134n2. Regra do Quociente`));a^n / a^m: %= simplify(%); print(txln);

NiMlXnF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzIuflJlZ3JhfmRvflF1b2NpZW50ZUc=NiMvKiYpJSJhRyUibkciIiIpJSJhRyUibUchIiIpJSJhRywmJSJuRyIiIiUibUchIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

10^9 / 10^6;

NiMiJSs1

x^14/x^13;

NiMlInhH

x^20 / x^23;

NiMqJiIiIiIiIiokKSUieEciIiQiIiIhIiI=

q^100 / q^10;

NiMqJCklInFHIiMhKiIiIg==

(y + 7)^3 / (y+7)^5;

NiMqJiIiIiIiIiokKSwmJSJ5RyIiIiIiKCIiIiIiIyIiIiEiIg==

print(cat(txln,`\134n3. A Todo n\303\272mero elevado a 0 \303\251 igual a 1`)); `a^0` = expand(a^0);print(txln);

NiMlZXJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzMufkF+VG9kb35ufGp5fGp2fGh2bWVyb35lbGV2YWRvfmF+MH58anl8anZ8aHZ+aWd1YWx+YX4xRw==NiMvJSRhXjBHIiIiNiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

13^0; 813^0; (-29)^0; (3/19)^0; (41239)^0; (3.14159)^0;

NiMiIiI=NiMiIiI=NiMiIiI=NiMiIiI=NiMiIiI=NiMkIiIiIiIh

print(cat(txln,`\134n4. Todo n\303\272mero elevado a 1 \303\251 igual a ele mesmo`));`a^1` =simplify(a^1); print(txln);

NiMlW3N8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzQuflRvZG9+bnxqeXxqdnxodm1lcm9+ZWxldmFkb35hfjF+fGp5fGp2fGh2fmlndWFsfmF+ZWxlfm1lc21vRw==NiMvJSRhXjFHJSJhRw==NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

13^1; 813^1; (-29)^1; (3/19)^1; (41239)^1; (3.14159)^1;

NiMiIzg=NiMiJDgpNiMhI0g=NiMjIiIkIiM+NiMiJlI3JQ==NiMkIidmVEohIiY=

print(cat(txln,`\134n5. Elevando duas pot\303\252ncias multiplicamos uma pela outra`)); (a^n)^m = a^(n*m) ; print(txln);

NiMlY3N8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzUufkVsZXZhbmRvfmR1YXN+cG90fGp5fGp2fGh2bmNpYXN+bXVsdGlwbGljYW1vc351bWF+cGVsYX5+b3V0cmFHNiMvKSklImFHJSJuRyUibUcpJSJhRyomJSJuRyIiIiUibUciIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

(x^2)^3;

NiMqJCklInhHIiInIiIi

(y^10)^5; `Compare com ...`, (y^10)*(y^5),` a for\303\247a da for\303\247a - produto de for\303\247as`;

NiMqJCklInlHIiNdIiIiNiUlMENvbXBhcmV+Y29tfi4uLkcqJCklInlHIiM6IiIiJUx+YX5mb3J8anl8anZ8aHZhfmRhfmZvcnxqeXxqdnxodmF+LX5wcm9kdXRvfmRlfmZvcnxqeXxqdnxodmFzRw==

((t - 7)^8 )^3 ;

NiMqJCksJiUidEciIiIiIighIiIiI0MiIiI=

print(cat(txln,`\134n6. A for\303\247a do produto`)); (a*b)^m = (a^n)*(b^m); print(txln);

NiMlYHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzYufkF+Zm9yfGp5fGp2fGh2YX5kb35wcm9kdXRvRw==NiMvKSomJSJhRyIiIiUiYkciIiIlIm1HKiYpJSJhRyUibkciIiIpJSJiRyUibUciIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

(3*x)^4;

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(8*y)^10;

NiMsJComIitDPXV0NSIiIiklInlHIiM1IiIiIiIi

print(cat(txln,`\134n7. A pot\303\252ncia do produto das for\303\247as `)); (a^n*b^m)^m = (a^(n*p))*(b^(m*p)); `(Isto \303\251 a combina\303\247\303\243o das regras de 1 a 6.)`; print(txln);

NiMlYXJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzcufkF+cG90fGp5fGp2fGh2bmNpYX5kb35wcm9kdXRvfmRhc35mb3J8anl8anZ8aHZhc35HNiMvKSomKSUiYUclIm5HIiIiKSUiYkclIm1HIiIiJSJtRyomKSUiYUcqJiUibkciIiIlInBHIiIiIiIiKSUiYkcqJiUibUciIiIlInBHIiIiIiIiNiMlUShJc3RvfnxqeXxqdnxodn5hfmNvbWJpbmF8anl8anZ8aHZ8anl8anZ8aHZvfmRhc35yZWdyYXN+ZGV+MX5hfjYuKUc=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

(x^3 * y^5) ^ 2;

NiMqJiklInhHIiInIiIiKSUieUciIzUiIiI=

(x^10 * y^20) ^ 5;

NiMqJiklInhHIiNdIiIiKSUieUciJCsiIiIi

(17 * M^7) ^ 2;

NiMsJComIiQqRyIiIiklIk1HIiM5IiIiIiIi

(3 * x^10 * y^7) ^ 4;

NiMsJCooIiMiKSIiIiklInhHIiNTIiIiKSUieUciI0ciIiIiIiI=

(a^2 * b^3 * c^4 * d^5 ) ^ 10;

NiMqKiklImFHIiM/IiIiKSUiYkciI0kiIiIpJSJjRyIjUyIiIiklImRHIiNdIiIi

print(cat(txln,`\134n8. A pot\303\252ncia do quociente`)); (a/b)^n =(a^n)/(b^n); print(txln);

NiMlZXF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzgufkF+cG90fGp5fGp2fGh2bmNpYX5kb35xdW9jaWVudGVHNiMvKSomJSJhRyIiIiUiYkchIiIlIm5HKiYpJSJhRyUibkciIiIpJSJiRyUibkchIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

(3/4)^3;

NiMjIiNGIiNr

(x/2)^5;

NiMsJComIiNLISIiKSUieEciIiYiIiIiIiI=

(a/b)^4;

NiMqJiklImFHIiIlIiIiKSUiYkciIiUhIiI=

(3/(x+5))^9;

NiMsJComIiYkbz4iIiIpLCYlInhHIiIiIiImIiIiIiIqISIiIiIi

print(cat(txln,`\134n9. A Pot\303\252ncia dos quocientes de for\303\247a`)); (a^n * b^m)^p = a^(n*p)*b^(m*p); `(Isto \303\251 a combina\303\247\303\243o das regras 2 e 8.)`; print(txln);

NiMlYXJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzkufkF+UG90fGp5fGp2fGh2bmNpYX5kb3N+cXVvY2llbnRlc35kZX5mb3J8anl8anZ8aHZHNiMvKSomKSUiYUclIm5HIiIiKSUiYkclIm1HIiIiJSJwRyomKSUiYUcqJiUibkciIiIlInBHIiIiIiIiKSUiYkcqJiUibUciIiIlInBHIiIiIiIiNiMlTihJc3RvfnxqeXxqdnxodn5hfmNvbWJpbmF8anl8anZ8aHZ8anl8anZ8aHZvfmRhc35yZWdyYXN+Mn5lfjguKUc=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

( x^3 / y^ 5)^2;

NiMqJiklInhHIiInIiIiKSUieUciIzUhIiI=

( 1 / pressure^2) ^ 3;

NiMqJiIiIiIiIiokKSUpcHJlc3N1cmVHIiInIiIiISIi

(x^10 / 3) ^ 5;

NiMsJComIiRWIyEiIiklInhHIiNdIiIiIiIi

Existem outras regras que envolvem expoentes negativos: print(cat(txln,`\134n10. Rec\303\255proca do expoente negativo`)); a^(-n) = 1/a^n; print(txln);

NiMlXnJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzEwLn5+UmVjfGp5fGp2fGh2cHJvY2F+ZG9+ZXhwb2VudGV+bmVnYXRpdm9HNiMvKSUiYUcsJCUibkchIiIqJiIiIiIiIiklImFHJSJuRyEiIg==NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

3^(-2);

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`Note :`, 9*3^(-2), 9*(1/9);

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x^(-13);

NiMqJiIiIiIiIiokKSUieEciIzgiIiIhIiI=

(-2)^(-3);

NiMjISIiIiIp

(-2)^(-4);

NiMjIiIiIiM7

(1/7)^(-2);

NiMiI1w=

print(cat(txln,`\134n11. Rec\303\255proco do rec\303\255proco \303\251 o original`)); 1/a^(-n) = a^n; print(txln);

NiMlZnJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzExLn5SZWN8anl8anZ8aHZwcm9jb35kb35yZWN8anl8anZ8aHZwcm9jb358anl8anZ8aHZ+b35vcmlnaW5hbEc=NiMvKiYiIiIiIiIpJSJhRywkJSJuRyEiIiEiIiklImFHJSJuRw==NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

1/(4^(-3));

NiMiI2s=

1/y^(-12);

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1/(3*x + 2)^(-5);

NiMqJCksJiomIiIkIiIiJSJ4RyIiIiIiIiIiIyIiIiIiJiIiIg==

`Observa\303\247\303\243o: Regras 10 e 11 podem ser usadas para o mesmo problema :`;x^(-4) / y^(-9);

NiMlYm9PYnNlcnZhfGp5fGp2fGh2fGp5fGp2fGh2bzp+UmVncmFzfjEwfmV+MTF+cG9kZW1+c2VyfnVzYWRhc35wYXJhfm9+bWVzbW9+cHJvYmxlbWF+Okc=NiMqJiklInhHIiIlISIiKSUieUciIioiIiI=

print(cat(txln,`\134n12. Reciproca da fra\303\247\303\243o \303\251 invertida`)); (a/b)^(-n) = (b/a)^n; print(txln);

NiMlYnJ8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzEyLn5SZWNpcHJvY2F+ZGF+ZnJhfGp5fGp2fGh2fGp5fGp2fGh2b358anl8anZ8aHZ+aW52ZXJ0aWRhRw==NiMvKSomJSJhRyIiIiUiYkchIiIsJCUibkchIiIpKiYlImJHIiIiJSJhRyEiIiUibkc=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

(2/131)^(-1);

NiMjIiRKIiIiIw==

(x/y)^(-1);

NiMqJiUieEchIiIlInlHIiIi

(1000/A)^(-1);

NiMsJComIiUrNSEiIiUiQUciIiIiIiI=

( (x+1) / (x+2) )^(-1);

NiMqJiwmJSJ4RyIiIiIiIiIiIiEiIiwmJSJ4RyIiIiIiIyIiIiIiIg==

( (x^2 + 3*x + 1) / (x^2 + 5*x + 2) )^(-1);

NiMqJiwoKiQpJSJ4RyIiIyIiIiIiIiomIiIkIiIiJSJ4RyIiIiIiIiIiIiIiIiEiIiwoKiQpJSJ4RyIiIyIiIiIiIiomIiImIiIiJSJ4RyIiIiIiIiIiIyIiIiIiIg==

(( A/B )^(-1))^(-1);

NiMqJiUiQUciIiIlIkJHISIi

Quando multiplicamos duas express\303\265es monomiais, os n\303\272meros podem ser multiplicados e as proprias vari\303\241veis podem ser multiplicadas - lembre-se do que aprendemos na regra #1 s\303\263 podemos multiplicar vari\303\241veis com a mesma base. Por exemplo, com duas express\303\265es com n\303\272meros , x e y - multiplicamos: - N\303\272mero por n\303\272mero - Coeficientes de x por coeficientes de x - Coeficientes de y por coeficientes de y (3 * x^7* y^2) * (4 * x^4 * y^9);

NiMsJCooIiM3IiIiKSUieEciIzYiIiIpJSJ5RyIjNiIiIiIiIg==

(5 * x^5 * y^2) * (11 * x^11 * y^(-3));

NiMsJCooIiNiIiIiKSUieEciIzsiIiIlInlHISIiIiIi

(13 * x^(-3)* y^(-8)) * (4 * x^(-2) * y^(-7));

NiMsJCooIiNfIiIiKSUieEciIiYhIiIpJSJ5RyIjOiEiIiIiIg==

(1440 * x^100* y^100) * (1200 * x^100 * y^(-100));

NiMsJComIigrIUc8IiIiKSUieEciJCsjIiIiIiIi

(33 * x^33* y^22) * (4 * x^44 * y^55);

NiMsJCooIiRLIiIiIiklInhHIiN4IiIiKSUieUciI3giIiIiIiI=

Quando multiplicamos precisamos usar a propriedade distributiva: `Propriedade Distributiva`; a*(b+c): % = expand(%);

NiMlOVByb3ByaWVkYWRlfkRpc3RyaWJ1dGl2YUc=NiMvKiYlImFHIiIiLCYlImJHIiIiJSJjRyIiIiIiIiwmKiYlImFHIiIiJSJiRyIiIiIiIiomJSJhRyIiIiUiY0ciIiIiIiI=

Ent\303\243o, significa que cada uma dessas multiplica\303\247\303\265es \303\251 um produto de mon\303\264mios. (12*x^10)*(14*x^3 - 1): % = expand(%);

NiMvLCQqKCIjNyIiIiklInhHIiM1IiIiLCYqJiIjOSIiIiklInhHIiIkIiIiIiIiIiIiISIiIiIiIiIiLCYqJiIkbyIiIiIpJSJ4RyIjOCIiIiIiIiomIiM3IiIiKSUieEciIzUiIiIhIiI=

(3*x^2)*(14*x + 21): % = expand(%);

NiMvLCQqKCIiJCIiIiklInhHIiIjIiIiLCYqJiIjOSIiIiUieEciIiIiIiIiI0AiIiIiIiIiIiIsJiomIiNVIiIiKSUieEciIiQiIiIiIiIqJiIjaiIiIiklInhHIiIjIiIiIiIi

(2*x^5)*(4*x^5 - x^2): % = expand(%);

NiMvLCQqKCIiIyIiIiklInhHIiImIiIiLCYqJiIiJSIiIiklInhHIiImIiIiIiIiKiQpJSJ4RyIiIyIiIiEiIiIiIiIiIiwmKiYiIikiIiIpJSJ4RyIjNSIiIiIiIiomIiIjIiIiKSUieEciIigiIiIhIiI=

(a^5*b^10)*(10*a^10*b^9 - a^8*b^7): % = expand(%);

NiMvKigpJSJhRyIiJiIiIiklImJHIiM1IiIiLCYqKCIjNSIiIiklImFHIiM1IiIiKSUiYkciIioiIiIiIiIqJiklImFHIiIpIiIiKSUiYkciIigiIiIhIiIiIiIsJiooIiM1IiIiKSUiYUciIzoiIiIpJSJiRyIjPiIiIiIiIiomKSUiYUciIzgiIiIpJSJiRyIjPCIiIiEiIg==

(20*a^5*b)*(13*a^4*b^9 - b^2): % = expand(%);

NiMvLCQqKiIjPyIiIiklImFHIiImIiIiJSJiRyIiIiwmKigiIzgiIiIpJSJhRyIiJSIiIiklImJHIiIqIiIiIiIiKiQpJSJiRyIiIyIiIiEiIiIiIiIiIiwmKigiJGcjIiIiKSUiYUciIioiIiIpJSJiRyIjNSIiIiIiIiooIiM/IiIiKSUiYUciIiYiIiIpJSJiRyIiJCIiIiEiIg==

Tamb\303\251m podemos resolver esses problemas com o Maple - e bem r\303\241pido. Tente primeiro \303\240 m\303\243o e depois veja se as suas respostas concordam com as do Maple e/ou se s\303\243o matematicamente equivalentes. (36 * x^4 * y^6 ) / (24 * x^5 * y^5);

NiMsJCoqIiIkIiIiIiIjISIiJSJ4RyEiIiUieUciIiIiIiI=

(75 * x^10 * y^3 ) / (24 * x * y^5);

NiMsJCoqIiNEIiIiIiIpISIiKSUieEciIioiIiIpJSJ5RyIiIyEiIiIiIg==

(100 * x * y ) / (45 * x^11 * y^9);

NiMsJCoqIiM/IiIiIiIqISIiKSUieEciIzUhIiIpJSJ5RyIiKSEiIiIiIg==

(32 * x^14 * y^6 ) / (40 * x^5 * y^5);

NiMsJCoqIiIlIiIiIiImISIiKSUieEciIioiIiIlInlHIiIiIiIi

(10 * x^(-4) * y^(-6) ) / (12 * x^5 * y^5);

NiMsJCoqIiImIiIiIiInISIiKSUieEciIiohIiIpJSJ5RyIjNiEiIiIiIg==

(33 * x^(-5) * y^6 ) / (22 * x^(-5) * y^(-5));

NiMsJCooIiIkIiIiIiIjISIiKSUieUciIzYiIiIiIiI=

(15 * x^4 * y^(-6) ) / (18 * x^(-4) * y^5);

NiMsJCoqIiImIiIiIiInISIiKSUieEciIikiIiIpJSJ5RyIjNiEiIiIiIg==

((6 * x^2 * y^3) / (30 * x^4 * y^5 ) ) ^(-1) ;

NiMsJCooIiImIiIiKSUieEciIiMiIiIpJSJ5RyIiIyIiIiIiIg==

((40*x^2*y^9) / (30*x^(-6)*y^8) )^(-2) ;

NiMsJCoqIiIqIiIiIiM7ISIiKSUieEciIzshIiIpJSJ5RyIiIyEiIiIiIg==

((27 * x^(-32) * y^19) / (24 * x^(-46) * y^8) )^(-4) ;

NiMsJCoqIiUnNCUiIiIiJWhsISIiKSUieEciI2MhIiIpJSJ5RyIjVyEiIiIiIg==