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# help

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Find all solutions to $$\sqrt[3]{x + 57} - \sqrt[3]{x - 57} = \sqrt[3]{6}.$$

Apr 14, 2019

#2
+25460
+3

Find all solutions to

$$\large \sqrt[3]{x + 57} - \sqrt[3]{x - 57} = \sqrt[3]{6}$$.

$$\begin{array}{|rcll|} \hline \sqrt[3]{x + 57} - \sqrt[3]{x - 57} &=& \sqrt[3]{6} \\ \left( \sqrt[3]{x + 57} - \sqrt[3]{x - 57} \right)^3 &=& 6 \\ (x + 57)-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2}-(x-57) &=& 6 \\ 114-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2} &=& 6 \\ 108-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2} &=& 0 \quad | \quad : 3\\ 36- \sqrt[3]{(x + 57)^2(x - 57)} + \sqrt[3]{(x + 57)(x - 57)^2} &=& 0 \\ 36- \sqrt[3]{(x^2 - 57^2)(x + 57)} + \sqrt[3]{(x^2 - 57^2)(x - 57)} &=& 0 \\ 36- \sqrt[3]{x^2 - 57^2} \underbrace{\left(\sqrt[3]{x + 57} - \sqrt[3]{x - 57}\right)}_{=\sqrt[3]{6}} &=& 0 \\ \sqrt[3]{x^2 - 57^2}\sqrt[3]{6} &=& 36 \\ (x^2 - 57^2)\cdot 6 &=& 36^3 \\ x^2 - 57^2 &=& 7776 \\ x^2 &=& 7776 +3249 \\ x^2 &=& 11025 \\ \mathbf{x} &\mathbf{=}& \mathbf{\pm 105} \\ \hline \end{array}$$

check $$x=105$$:

$$\begin{array}{rcll} \sqrt[3]{105 + 57} - \sqrt[3]{105 - 57} &\overset{?}{=}& \sqrt[3]{6} \\ \sqrt[3]{162} - \sqrt[3]{48} &\overset{?}{=}& 1.81712059283 \\ 5.45136177850 - 3.63424118566 &\overset{?}{=}& 1.81712059283 \\ 1.81712059283 & = & 1.81712059283 \quad \checkmark \\ \end{array}$$

check $$x=-105$$:

$$\begin{array}{rcll} \sqrt[3]{-105 + 57} - \sqrt[3]{-105 - 57} &\overset{?}{=}& \sqrt[3]{6} \\ \sqrt[3]{-48} - \sqrt[3]{-162} &\overset{?}{=}& 1.81712059283 \\ - 3.63424118566 -(-5.45136177850 ) &\overset{?}{=}& 1.81712059283 \\ - 3.63424118566 + 5.45136177850 &\overset{?}{=}& 1.81712059283 \\ 1.81712059283 & = & 1.81712059283 \quad \checkmark \\ \end{array}$$

Apr 15, 2019
edited by heureka  Apr 16, 2019

#1
+111431
+1

WolframAlpha shows a solution of x = 105

Maybe a substitution could solve this??? [ But.....I'm not sure ]

Anyone know an algebraic solution  ????

Apr 14, 2019
#2
+25460
+3

Find all solutions to

$$\large \sqrt[3]{x + 57} - \sqrt[3]{x - 57} = \sqrt[3]{6}$$.

$$\begin{array}{|rcll|} \hline \sqrt[3]{x + 57} - \sqrt[3]{x - 57} &=& \sqrt[3]{6} \\ \left( \sqrt[3]{x + 57} - \sqrt[3]{x - 57} \right)^3 &=& 6 \\ (x + 57)-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2}-(x-57) &=& 6 \\ 114-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2} &=& 6 \\ 108-3 \sqrt[3]{(x + 57)^2(x - 57)} + 3 \sqrt[3]{(x + 57)(x - 57)^2} &=& 0 \quad | \quad : 3\\ 36- \sqrt[3]{(x + 57)^2(x - 57)} + \sqrt[3]{(x + 57)(x - 57)^2} &=& 0 \\ 36- \sqrt[3]{(x^2 - 57^2)(x + 57)} + \sqrt[3]{(x^2 - 57^2)(x - 57)} &=& 0 \\ 36- \sqrt[3]{x^2 - 57^2} \underbrace{\left(\sqrt[3]{x + 57} - \sqrt[3]{x - 57}\right)}_{=\sqrt[3]{6}} &=& 0 \\ \sqrt[3]{x^2 - 57^2}\sqrt[3]{6} &=& 36 \\ (x^2 - 57^2)\cdot 6 &=& 36^3 \\ x^2 - 57^2 &=& 7776 \\ x^2 &=& 7776 +3249 \\ x^2 &=& 11025 \\ \mathbf{x} &\mathbf{=}& \mathbf{\pm 105} \\ \hline \end{array}$$

check $$x=105$$:

$$\begin{array}{rcll} \sqrt[3]{105 + 57} - \sqrt[3]{105 - 57} &\overset{?}{=}& \sqrt[3]{6} \\ \sqrt[3]{162} - \sqrt[3]{48} &\overset{?}{=}& 1.81712059283 \\ 5.45136177850 - 3.63424118566 &\overset{?}{=}& 1.81712059283 \\ 1.81712059283 & = & 1.81712059283 \quad \checkmark \\ \end{array}$$

check $$x=-105$$:

$$\begin{array}{rcll} \sqrt[3]{-105 + 57} - \sqrt[3]{-105 - 57} &\overset{?}{=}& \sqrt[3]{6} \\ \sqrt[3]{-48} - \sqrt[3]{-162} &\overset{?}{=}& 1.81712059283 \\ - 3.63424118566 -(-5.45136177850 ) &\overset{?}{=}& 1.81712059283 \\ - 3.63424118566 + 5.45136177850 &\overset{?}{=}& 1.81712059283 \\ 1.81712059283 & = & 1.81712059283 \quad \checkmark \\ \end{array}$$

heureka Apr 15, 2019
edited by heureka  Apr 16, 2019
#3
+111431
+3

Wow!!!!....that's impressive, Heureka!!!

Definitely one for my "Watchlist"

CPhill  Apr 15, 2019
#4
+25460
+3

Thank you, CPhill !

heureka  Apr 16, 2019