# How do you simplify (root3(6)*root4(6))^12?

Mar 19, 2017

${\left(\sqrt[3]{6} \cdot \sqrt[4]{6}\right)}^{12} = {6}^{7} = 279936$

#### Explanation:

We need the rules:

• $\sqrt[m]{x} = {x}^{\frac{1}{m}} \text{ "" } \textcolor{red}{\star}$
• ${x}^{a} \cdot {x}^{b} = {x}^{a + b} \text{ "" } \textcolor{g r e e n}{\star}$
• ${\left({x}^{c}\right)}^{d} = {x}^{c d} \text{ "" } \textcolor{b l u e}{\star}$

Using $\textcolor{red}{\star}$, we see that:

${\left(\sqrt[3]{6} \cdot \sqrt[4]{6}\right)}^{12} = {\left({6}^{\frac{1}{3}} \cdot {6}^{\frac{1}{4}}\right)}^{12}$

Now using $\textcolor{g r e e n}{\star}$, this becomes:

${\left({6}^{\frac{1}{3}} \cdot {6}^{\frac{1}{4}}\right)}^{12} = {\left({6}^{\frac{1}{3} + \frac{1}{4}}\right)}^{12}$

Note that $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

${\left({6}^{\frac{1}{3} + \frac{1}{4}}\right)}^{12} = {\left({6}^{\frac{7}{12}}\right)}^{12}$

Now using $\textcolor{b l u e}{\star}$, we multiply the exponents:

${\left({6}^{\frac{7}{12}}\right)}^{12} = {6}^{\frac{7}{12} \times 12}$

And we see that $\frac{7}{12} \times 12 = 7$:

${6}^{\frac{7}{12} \times 12} = {6}^{7}$

All of which we did without a calculator! For an expanded value, we could plug in ${6}^{7}$ into a calculator to see that ${6}^{7} = 279936$.