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# Can you help? Thanks so much

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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
+25644
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If, $$\left(\sqrt[3]{7}\right)^{5x}=14$$ what is the value of $$\left(\sqrt[3]{7}\right)^{10x-6}$$

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

Jul 13, 2020
#2
0

Thanks so much!

Jul 13, 2020
#3
+111983
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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