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Simplify
[tex]\[ \sqrt{\sqrt{162 p^{10} q^8}} \][/tex]

Sagot :

GinnyAnsweridn
Para simplificar la expresión
[tex]\[ \sqrt{\sqrt{162 p^{10} q^8}} \][/tex]
vamos a descomponer y simplificar paso a paso:

1. Identificar la estructura de la expresión:
La expresión original es:
[tex]\[ \sqrt{\sqrt{162 p^{10} q^8}} \][/tex]
Podemos reescribirlo usando exponentes fraccionarios para mayor claridad:
[tex]\[ \left(162 p^{10} q^8\right)^{\frac{1}{4}} \][/tex]

2. Descomponer 162 en sus factores primos:
Sabemos que:
[tex]\[ 162 = 2 \cdot 81 = 2 \cdot 3^4 \][/tex]
Entonces, la expresión ahora es:
[tex]\[ \left(2 \cdot 3^4 \cdot p^{10} \cdot q^8 \right)^{\frac{1}{4}} \][/tex]

3. Distribuir el exponente [tex]\(\frac{1}{4}\)[/tex] a cada factor dentro de los paréntesis:
Aplicamos la propiedad de los exponentes [tex]\((ab)^n = a^n b^n\)[/tex]:
[tex]\[ \left(2^{1} \cdot 3^4 \cdot p^{10} \cdot q^8\right)^{\frac{1}{4}} = 2^{\frac{1}{4}} \cdot 3^{4 \cdot \frac{1}{4}} \cdot (p^{10})^{\frac{1}{4}} \cdot (q^8)^{\frac{1}{4}} \][/tex]

4. Simplificar cada término individualmente:
- Para [tex]\(3^{4 \cdot \frac{1}{4}}\)[/tex]:
[tex]\[ 3^{4 \cdot \frac{1}{4}} = 3^1 = 3 \][/tex]
- Para [tex]\((p^{10})^{\frac{1}{4}}\)[/tex]:
[tex]\[ (p^{10})^{\frac{1}{4}} = p^{10 \cdot \frac{1}{4}} = p^{\frac{10}{4}} = p^{\frac{5}{2}} \][/tex]
- Para [tex]\((q^8)^{\frac{1}{4}}\)[/tex]:
[tex]\[ (q^8)^{\frac{1}{4}} = q^{8 \cdot \frac{1}{4}} = q^2 \][/tex]

5. Combinar todos los términos simplificados:
Entonces tenemos:
[tex]\[ \sqrt{\sqrt{162 p^{10} q^8}} = 2^{\frac{1}{4}} \cdot 3 \cdot p^{\frac{5}{2}} \cdot q^2 \][/tex]

6. Reescribir de forma más compacta:
La expresión puede reescribirse para reflejar mejor la simplificación:
[tex]\[ 3 \cdot 2^{\frac{1}{4}} \cdot \left(p^{10} q^8\right)^{\frac{1}{4}} \][/tex]

Por lo tanto, la forma simplificada de
[tex]\[ \sqrt{\sqrt{162 p^{10} q^8}} \][/tex]
es:
[tex]\[ 3 \cdot 2^{\frac{1}{4}} \cdot (p^{10} q^8)^{\frac{1}{4}} \][/tex]
Hi1315idn

Answer:

[tex]3 \times \sqrt[4]{2} \times p^{5/2} \times q^2[/tex]

Step-by-step explanation:

Let's simplify [tex]\sqrt{\sqrt{162 p^{10} q^8}}[/tex] in a simpler way:

1. Simplify inside the inner square root first:

  [tex]\sqrt{162 p^{10} q^8}[/tex]

  Notice:

[tex]162 = 81 \times 2 = 9^2 \times 2[/tex]

[tex]p^{10} = (p^5)^2[/tex]

[tex]q^8 = (q^4)^2[/tex]

  So:

[tex]\sqrt{162 p^{10} q^8} = \sqrt{(9^2 \times 2) \times (p^5)^2 \times (q^4)^2}\\\\ \text{When you take the square root:}\\\\ \sqrt{9^2} \times \sqrt{2} \times \sqrt{(p^5)^2} \times \sqrt{(q^4)^2}[/tex]

  Simplify:

[tex]9 \times \sqrt{2} \times p^5 \times q^4[/tex]

2. Now simplify the outer square root:

[tex]\sqrt{9 \sqrt{2} p^5 q^4}[/tex]

[tex]\text{Notice:}\\\\ \sqrt{9} = 3\\\\ \sqrt{p^5} = p^{5/2}\\\\ \sqrt{q^4} = q^2[/tex]

[tex]So:\\ 3 \times \sqrt[4]{2} \times p^{5/2} \times q^2[/tex]

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