Página — Matematica 6° Secundaria (pág. 56)

Propiedades de la potenciación del sistema de los números naturales

• Potencia de un producto.Ejemplo:

\begin{aligned}&(3\times2)^{2}=(3\times2)\times(3\times2)=6\times6=36\\&3^{2}\times2^{2}=3\times3\times2\times2=9\times4=36\\&\mathbf{L}\mathbf{e}\mathbf{g}\mathbf{o}(3\times2)^{2}=3^{2}\times2^{2}\mathbf{o}\mathbf{t}\mathbf{m}\mathbf{b}\mathbf{i}\mathbf{e}\mathbf{n}3^{2}\times2^{2}=(3\times2)^{2}\\ \end{aligned}

La potencia de un producto es igual al producto de las potencias que se obtienen al elevar cada factor al exponente dado.

• Producto de potencias de igual base.Ejemplo:

\begin{array}{l}3^{2}\times3^{1}=\\3^{2}\times3^{1}=3\times3\times3=27\\(3)^{2+1}=3^{3}=3\times3\times3=27,\\\text{Entoces:}3^{2}\times3^{1}=(3)^{2+1}\text{o tambien}(3)^{2+1}=3^{2}\times3^{1}\end{array}

EI producto de potencias de igual base se calcula dejando la misma base y sumando los exponentes.

• División de potencias con igual base.

\begin{aligned}&\mathrm{Ejemplo}:\\ &\frac{4^{5}}{4^{3}}=\\ &\frac{4^{5}}{4^{3}}=\frac{4\times4\times4\times4\times4}{4\times4\times4}=4\times4=16\\ \end{aligned}

Para dividir dos potencias con la misma base se escribe la misma base y se restan los exponentes.

• Potencia de una potencia.Ejemplo:

\begin{array}{l}(2^{2})^{3}=(2^{2})^{3=}(2\times2)^{3=}4^{3}=4\times4\times4=64\\(2)^{2\times3}=2^{6}=2\times2\times2\times2\times2=64\\\operatorname{Luego},(2^{2})^{3}=(2)^{2\times3}\mathrm{o}(2)^{2\times3}=(2^{2})^{3}\end{array}

Para hallar la potencia de una potencia, se escribe la misma base y se multiplican los exponentes.

• Todo número natural elevado al exponente 1 es igual al mismo número natural.

Ejemplo:4^{1}=4

• Todo número Natural elevado al exponente O es igual a 1.

Ejemplo:

5^{0}=1

5^{0}=(5)^{3-3} el cero es equivalente a 3-3 o 2-2

5^{0}=\frac{5^{3}}{5^{3}} División de potencias de la misma base.5^{0}=1 el cociente de un número por él mismo es igual a 1.

La radicación es una operación inversa a la potenciación, se aplica cuando conociendo el exponente y la potencia, se desea conocer la base.

Por ejemplo:

\sqrt[4]{81}=3\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad388888888\quad\quad838}888888888888}8888=3333888888=3399999999=99998999=89999898998999899=9888898898999989988989998=999999998889899989999988998899999899999998899999899999999

El gráfico siguiente muestra los términos de la radicación:\begin{array}{c}\mid\mid\\mid\mid\\mid\mid\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\\mid\mid\\mid\ \\mid\\mid\\mid\ \\ \mid\ \mid\mid\mid\ \\ \mid\mid\ \\ \\ \mid\mid\mid\mid\mid\mid\ \\ \mid\ \\ \mid\ \mid\mid\mid\ \mid\mid\ \mid\mid\ \mid\mid\ \\ \\ \\ \mid\mid\mid\ \mid\ \mid\mid\mid\ \mid\mid\ \\ \mid\ \mid\mid\ \mid\mid\mid\mid\mid\mid\mid\ \mid\ \\ \\ \mid\mid\mid\mid\mid\mid\ \mid\ \mid\mid\mid\ \mid\ \mid\mid\mid\mid\mid\mid\ \mid\ \mid\mid\mid\ \mid\mid\mid\mid\ \mid\mid\mid\mid\mid\mid\mid\ \mid\mid\mid\mid\mid\mid\mid\mid\ \\ \mid\mid\mid\mid\mid\mid\ \mid\mid\mid\mid\mid\mid\mid\ \mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid$

$\begin{array}{c}\mid\mid\\\mid\mid\\\mid\mid\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\\\mid\mid\\\mid\\ \\\mid\\\mid\\\mid\\ \\ \mid\\ \mid\mid\mid\\ \\ \mid\mid\\ \\ \\ \mid\mid\mid\mid\mid\mid\\ \\ \mid\\ \\ \mid\\ \mid\mid\mid\\ \mid\mid\\ \mid\mid\\ \mid\mid\\ \\ \\ \\ \mid\mid\mid\\ \mid\\ \mid\mid\mid\\ \mid\mid\\ \\ \mid\\ \mid\mid\\ \mid\mid\mid\mid\mid\mid\mid\\ \mid\\ \\ \\ \mid\mid\mid\mid\mid\mid\\ \mid\\ \mid\mid\mid\\ \mid\\ \mid\mid\mid\mid\mid\mid\\ \mid\\ \mid\mid\mid\\ \mid\mid\mid\mid\\ \mid\mid\mid\mid\mid\mid\mid\\ \mid\mid\mid\mid\mid\mid\mid\mid\\ \\ \mid\mid\mid\mid\mid\mid\\ \mid\mid\mid\mid\mid\mid\mid\\ \mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid\mid

Si la raíz tiene índice 2 se lee: "raiz cuadrada".Por lo general el índice 2 de la raíz cuadrada no se escribe.

\sqrt{16}=4 (la raíz cuadrada de 16 es igual a 4).

Sí la raíz tiene índice 3 se lee: "raíz cubica"Si la raíz tiene índice 4 se lee: "raíz cuarta", etc.

Propiedades de la radicación en el sistema de los números Naturales

•La raz de un producto.

Ejemplo:

\sqrt{25\times64}=\sqrt{25}\times\sqrt{64}=5\times8=40

Secundaria Activa // Ministerio de Educación Nacional

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Unidad1.Construyo sistemas de números