Respuestas
Respuesta:
Al resolver se reduce a su forma más simple las expresiones algebraicas:
1. ∛8x³ = 2x
2. \sqrt[3]{8}.\sqrt[3]{x^{4}}=2x^{\frac{4}{3}}
3
8
.
3
x
4
=2x
3
4
3. \frac{\sqrt[3]{8x^{2}}}{\sqrt[3]{27x^{5}}}=\frac{2}{3}.\frac{1}{x}
3
27x
5
3
8x
2
=
3
2
.
x
1
4. \sqrt[3]{\sqrt{a} }=\sqrt[6]{a}
3
a
=
6
a
Explicación paso a paso:
1. ∛8x³ =
Aplicar propiedad de raíces;
\begin{gathered}\sqrt[n]{a}.\sqrt[n]{b}=\sqrt[n]{a.b} \\\sqrt[n]{x^{m} }=x^{\frac{m}{n} }\end{gathered}
n
a
.
n
b
=
n
a.b
n
x
m
=x
n
m
=∛8 • ∛x
= 2 •∛x³
= 2 •x
2. ∛8 • ∛x⁴ =
Aplicar propiedad de raíces;
\begin{gathered}\sqrt[n]{a}.\sqrt[n]{b}=\sqrt[n]{a.b} \\\sqrt[n]{x^{m} }=x^{\frac{m}{n} }\end{gathered}
n
a
.
n
b
=
n
a.b
n
x
m
=x
n
m
\sqrt[3]{8}.\sqrt[3]{x^{4} } =2.x^{\frac{4}{3}}
3
8
.
3
x
4
=2.x
3
4
3. \frac{\sqrt[3]{8x^{2}}}{\sqrt[3]{27x^{5}}}=
3
27x
5
3
8x
2
=
Aplicar propiedad de raíces;
\begin{gathered}\sqrt[n]{a}.\sqrt[n]{b}=\sqrt[n]{a.b} \\\sqrt[n]{x^{m} }=x^{\frac{m}{n} } \\\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\end{gathered}
n
a
.
n
b
=
n
a.b
n
x
m
=x
n
m
n
b
n
a
=
n
b
a
\begin{gathered}\frac{\sqrt[3]{8x^{2}}}{\sqrt[3]{27x^{5}}}\\=\frac{\sqrt[3]{8}.\sqrt[3]{x^{2}}}{\sqrt[3]{27}.\sqrt[3]{x^{5}}} \\=\frac{2.\sqrt[3]{x^{2}}}{8.\sqrt[3]{x^{5}}} \\=\frac{2}{8}.\sqrt[3]{\frac{x^{2} }{x^{5}}} \\=\frac{2}{8}.\sqrt[3]{x^{2-5}} \\=\frac{2}{8}.\sqrt[3]{x^{-3}}\\=\frac{2}{8}.x^{\frac{-3}{3} }\\=\frac{2}{3}.\frac{1}{x}\end{gathered}
3
27x
5
3
8x
2
=
3
27
.
3
x
5
3
8
.
3
x
2
=
8.
3
x
5
2.
3
x
2
=
8
2
.
3
x
5
x
2
=
8
2
.
3
x
2−5
=
8
2
.
3
x
−3
=
8
2
.x
3
−3
=
3
2
.
x
1
4. \sqrt[3]{\sqrt{a} } =
3
a
=
Aplicar propiedad e raíces;
\sqrt[n]{\sqrt[m]{x} } = \sqrt[n.m]{x}
n
m
x
=
n.m
x
\begin{gathered}\sqrt[3]{\sqrt{a} }= \sqrt[3.2]{a}\\=\sqrt[6]{a}\end{gathered}
3
a
=
3.2
a
=
6
a