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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.

 Jul 13, 2020
 #1
avatar+25555 
+2

If, \(\left(\sqrt[3]{7}\right)^{5x}=14\) what is the value of \(\left(\sqrt[3]{7}\right)^{10x-6}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{ \left(\sqrt[3]{7}\right)^{5x}} &=& \mathbf{14} \quad | \quad \text{square both sides} \\ \left( \left(\sqrt[3]{7}\right)^{5x} \right)^2 &=& 14^2 \\ \left(\sqrt[3]{7}\right)^{2*5x} &=& 196 \\ \left(\sqrt[3]{7}\right)^{10x} &=& 196 \quad | \quad : \left(\sqrt[3]{7}\right)^6 \\\\ \dfrac{ \left(\sqrt[3]{7}\right)^{10x} } {\left(\sqrt[3]{7}\right)^6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x}*\left(\sqrt[3]{7}\right)^{-6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{\left(\sqrt[3]{7}\right)^6} \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{ 7^{\frac{6}{3}} } \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{7^2 } \\\\ \left(\sqrt[3]{7}\right)^{10x-6} &=& \dfrac{196}{49} \\\\ \mathbf{\left(\sqrt[3]{7}\right)^{10x-6}} &=& \mathbf{4} \\ \hline \end{array}\)

 

laugh

 Jul 13, 2020
 #2
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Thanks so much!

 Jul 13, 2020
 #3
avatar+110727 
+2

Thanks Heureka, 

Here is a slightly different approach.

 

\(\left(\sqrt[3]{7}\right)^{5x}=14\)             find       \(\left(\sqrt[3]{7}\right)^{10x-6} \)

 

Let    \(y=\sqrt[3]{7}\)

 

\(y^{5x}=14\qquad \qquad find \qquad y^{10x-6}\\ y^{10x-6}=y^{5x}*y^{5x}*y^{-6}\\ y^{10x-6}=14*14*y^{-6}\\ (\sqrt[3]{7})^{10x-6}=196*(\sqrt[3]{7})^{-6}\\ (\sqrt[3]{7})^{10x-6}=196*7^{-2}\\ (\sqrt[3]{7})^{10x-6}=4\)

 Jul 14, 2020

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