Question

Simplify and leave in radical form.
https://media.glynlyon.com/g_alg02_ccss_2013/9/q13143.gif

Answer 1

\(\Large \bf \sqrt{3}\cdot \sqrt[3]{2}\implies \sqrt[2]{3}\cdot \sqrt[3]{2}\implies 3^{\frac{1}{2}}\cdot 2^{\frac{1}{3}}\)

fair?

Answer 2

Huh?

Answer 3

\(\huge \bf a^{\frac{{\color{blue} n}}{{\color{red} m}}} = \sqrt[{\color{red} m}]{a^{\color{blue} n}} \qquad \qquad

\sqrt[{\color{red} m}]{a^{\color{blue} n}}=a^{\frac{{\color{blue} n}}{{\color{red} m}}} \) <--- assuming you've covered that by now

Answer 4

ok

Answer 5

so \(\Large \bf \Large \bf \sqrt{3}\cdot \sqrt[3]{2}\implies \sqrt[2]{3}\cdot \sqrt[3]{2}\implies 3^{\frac{1}{2}}\cdot 2^{\frac{1}{3}}\) follow that thus far?

Answer 6

yes

Answer 7

ok so

can we say that \( \bf \cfrac{3}{6}\iff \cfrac{1}{2}\quad and\quad \cfrac{2}{6}\iff \cfrac{1}{3}\quad ?\)

Answer 8

yes?

Answer 9

hehh.... can we? are those fractions not equivalent? what do you think?

Answer 10

yes

Answer 11

ok.... then we could then say that \(\Large {\bf \sqrt{3}\cdot \sqrt[3]{2}\implies \sqrt[2]{3}\cdot \sqrt[3]{2}\implies 3^{\frac{1}{2}}\cdot 2^{\frac{1}{3}}
\\ \quad \\
3^{\frac{1}{2}\to {\color{brown}{ \frac{3}{6}}}}\cdot 2^{\frac{1}{3}\to {\color{brown}{ \frac{2}{6}}}}\implies 3^{{\color{brown}{ \frac{3}{6}}}}\cdot 2^{ {\color{brown}{ \frac{2}{6}}}}
}\)

fair?

Answer 12

fair, but how would I write the answer

Answer 13

\(\Large \bf {
\sqrt{3}\cdot \sqrt[3]{2}\implies \sqrt[2]{3}\cdot \sqrt[3]{2}\implies 3^{\frac{1}{2}}\cdot 2^{\frac{1}{3}}
\\ \quad \\
3^{\frac{1}{2}\to {\color{brown}{ \frac{3}{6}}}}\cdot 2^{\frac{1}{3}\to {\color{brown}{ \frac{2}{6}}}}\implies 3^{{\color{brown}{ \frac{3}{6}}}}\cdot 2^{ {\color{brown}{ \frac{2}{6}}}}\implies \sqrt[6]{3^3}\cdot \sqrt[6]{2^2}
\\ \quad \\
\sqrt[6]{3^3\cdot 2^2}\implies ?
}\)

see, once the root matches, then you can simply multiply the radicands and combine

Answer 14

27,4