Find-the-value-of-x-if-x-x-6-729-

Question Number 5385 by sanusihammed last updated on 12/May/16

F i n d t h e v a l u e o f x i f

\sqrt{x^{x^{6}}} = 729

Answered by prakash jain last updated on 12/May/16

\sqrt{x^{x^{6}}} = 729 = 3^{6}

x^{x^{6}} = 3^{12}

x^{x^{6}} = {(3^{2})}^{6}

x^{x} = 9

If the question was cube root on LHS

\sqrt[3]{x^{x^{6}}} = 729

x^{x^{6}} = 3^{18}

x^{x} = 3^{3} \Rightarrow x = 3

Commented by Rasheed Soomro last updated on 16/May/16

x^{x^{6}} \overset{?}{=} {(x^{x})}^{6}

Or x^{x^{6}} = x^{(x^{6})} ?

x^{x^{6}} = {(3^{2})}^{6} \overset{?}{\Rightarrow} x^{x} = 9

Answered by Rasheed Soomro last updated on 15/May/16

\sqrt{x^{x^{6}}} = 729 \Rightarrow x^{\frac{1}{2} x^{6}} = 729

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To write 729 in the base of x

x^{} = 729

\log_{3} x = \log_{3} 729 = 6

= \frac{6}{\log_{3} x}

729 = x^{\frac{6}{\log_{3} x}}

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x^{\frac{1}{2} x^{6}} = x^{\frac{6}{\log_{3} x}} \Rightarrow \frac{1}{2} x^{6} = \frac{6}{\log_{3} x}

\frac{1}{2} x^{6} \log_{3} x = 6

x^{6} \log_{3} x = 12

x^{6} = \frac{3 \times 4}{\log_{3} x} = \frac{\log_{3} 27 \times 4}{\log_{3} x} = \frac{\log_{3} 27^{4}}{\log_{3} x} = \log_{x} 27^{4}

x^{6} = \log_{x} 27^{4}

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