How do you simplify the following problem?

$\sqrt[3]{24} + \sqrt[3]{81}$

Mar 9, 2018

$\text{The answer is:} \setminus q \quad \setminus q \quad \setminus q \quad \sqrt[3]{24} + \sqrt[3]{81} \setminus q \quad = \setminus q \quad 5 \sqrt[3]{3} .$

Explanation:

$\text{We can do this using a property or two of radicals ... }$

$\text{We are given ... }$

$\setminus q \quad \sqrt[3]{24} + \sqrt[3]{81} \setminus q \quad = \setminus q \quad \sqrt[3]{8 \cdot 3} + \sqrt[3]{27 \cdot 3}$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now insert mental emphasis} \setminus q \quad \rightarrow}$

$= \setminus \left[\sqrt[3]{\setminus {\overbrace{\textcolor{red}{8}}}^{\text{ " 8 \ "is a perfect cube" } cdot 3 } \ ] \qquad + \qquad [ root{3}{\overbrace{ color{red}{27} }^ { " " 27 \ "is a perfect cube}} \cdot 3} \setminus \setminus\right]$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now use a basic property of radicals:} \setminus q \quad \sqrt[n]{a \cdot b} \setminus = \setminus \sqrt[n]{a} \cdot \sqrt[n]{b} \setminus \quad \rightarrow}$

$= \setminus \left[\sqrt[3]{\setminus {\overbrace{\textcolor{red}{8}}}^{\text{ " 8 \ "is a perfect cube" } } cdot root{3}{3} ] \quad + \quad [ root{3}{\overbrace{ color{red}{27} }^ { " " 27 \ "is a perfect cube}}} \cdot \sqrt[3]{3}\right]$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now remove some of the inner emphasis} \setminus q \quad \rightarrow}$

$= \setminus \left[\sqrt[3]{\textcolor{red}{8}} \cdot \sqrt[3]{3} \setminus\right] \setminus \quad + \setminus \quad \left[\sqrt[3]{\textcolor{red}{27}} \cdot \sqrt[3]{3} \setminus\right]$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now use basic numerical facts (about 8, and 27)} \setminus q \quad \rightarrow}$

$= \setminus \left[\textcolor{red}{2} \cdot \sqrt[3]{3} \setminus\right] \setminus \quad + \setminus \quad \left[\textcolor{red}{3} \cdot \sqrt[3]{3} \setminus\right]$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now reset to a new emphasis} \setminus q \quad \rightarrow}$

$= \setminus \left[2 \cdot \textcolor{red}{\sqrt[3]{3}} \setminus\right] \setminus \quad + \setminus \quad \left[3 \cdot \textcolor{red}{\sqrt[3]{3}} \setminus\right]$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now factor out a common factor:} \setminus q \quad \textcolor{red}{\sqrt[3]{3}} \setminus q \quad \rightarrow}$

$= \setminus \left[2 + 3\right] \cdot \textcolor{red}{\sqrt[3]{3}}$

$\setminus q \quad \setminus \textcolor{b l u e}{\text{now basic arithmetic, then remove emphasis & finish !!! } \setminus \quad \rightarrow}$

$= \setminus \left[5\right] \cdot \textcolor{red}{\sqrt[3]{3}}$

$= \setminus 5 \sqrt[3]{3} .$

$\text{This is our answer !!}$

$\text{So, summarizing, we have:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \sqrt[3]{24} + \sqrt[3]{81} \setminus q \quad = \setminus q \quad 5 \sqrt[3]{3} .$