If , \(\left(\sqrt[3]{7}\right)^{5x}=14\)what is the value of \(\left(\sqrt[3]{7}\right)^{10x-6}\) I don't really understand this problem. Thanks.
+26367 If, \(\left(\sqrt[3]{7}\right)^{5x}=14\) what is the value of \(\left(\sqrt[3]{7}\right)^{10x-6}\)
\(\begin{array}{|rcll|} \hline \mathbf{ \left(\sqrt[3]{7}\right)^{5x}} &=& \mathbf{14} \quad | \quad \text{square both sides} \\ \left( \left(\sqrt[3]{7}\right)^{5x} \right)^2 &=& 14^2 \\ \left(\sqrt[3]{7}\right)^{2*5x} &=& 196 \\ \left(\sqrt[3]{7}\right)^{10x} &=& 196 \quad | \quad : \left(\sqrt[3]{7}\right)^6 \\\\ \dfrac{ \left(\sqrt[3]{7}\right)^{10x} } {\left(\sqrt[3]{7}\right)^6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x}*\left(\sqrt[3]{7}\right)^{-6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{ 7^{\frac{6}{3}} } \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{7^2 } \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{49} \\\\ \mathbf{\left(\sqrt[3]{7}\right)^{10x-6}} &=& \mathbf{4} \\ \hline \end{array}\)
+118608 Thanks Heureka,
Here is a slightly different approach.
\(\left(\sqrt[3]{7}\right)^{5x}=14\) find \(\left(\sqrt[3]{7}\right)^{10x-6} \)
Let \(y=\sqrt[3]{7}\)
\(y^{5x}=14\qquad \qquad find \qquad y^{10x-6}\\ y^{10x-6}=y^{5x}*y^{5x}*y^{-6}\\ y^{10x-6}=14*14*y^{-6}\\ (\sqrt[3]{7})^{10x-6}=196*(\sqrt[3]{7})^{-6}\\ (\sqrt[3]{7})^{10x-6}=196*7^{-2}\\ (\sqrt[3]{7})^{10x-6}=4\)
