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The 100th root of ${{10}^{\left( {{10}^{10}} \right)}}$ is:
c.${{\left( \sqrt{10} \right)}^{{{\left( \sqrt{10} \right)}^{10}}}}$
d.$10{{\left( \sqrt{10} \right)}^{\sqrt{10}}}$

Last updated date: 12th Jul 2024
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Hint: Let the required value is $x$. If we multiply $x$ for 100 times we will get ${{10}^{\left( {{10}^{10}} \right)}}$.
Form an equation according to this condition and solve the value of $x$.

Complete step by step answer:
Let the number is $x$.
Therefore the 100th root of ${{10}^{\left( {{10}^{10}} \right)}}$ is $x$.
That means if we multiply $x$ for 100 times we will get ${{10}^{\left( {{10}^{10}} \right)}}$.
Or we can say if we take $x$ to the power 100 we will get ${{10}^{\left( {{10}^{10}} \right)}}$. An expression that represents repeated multiplication of the same factor is called a power. Or we can say the power of a number says how many times to use the number in a multiplication.
Hence, ${{x}^{100}}={{10}^{\left( {{10}^{10}} \right)}}$
From this equation we have to find out the value of $x$.
Therefore we have,
${{x}^{100}}={{10}^{\left( {{10}^{10}} \right)}}$
Now we can take the power of $x$ that is 100 from our left hand side to right hand side,
  & \Rightarrow x={{\left( {{10}^{\left( {{10}^{10}} \right)}} \right)}^{\dfrac{1}{100}}} \\
 & \Rightarrow x={{10}^{\dfrac{{{10}^{10}}}{100}}} \\
We can write 100 as 10 multiplied by 10. Therefore,
  & \Rightarrow x={{10}^{\dfrac{{{10}^{10}}}{10\times 10}}} \\
 & \Rightarrow x={{10}^{\dfrac{{{10}^{10}}}{{{10}^{2}}}}} \\
 & \Rightarrow x={{10}^{{{10}^{10-2}}}} \\
 & \Rightarrow x={{10}^{{{10}^{8}}}} \\
Therefore, the 100th root of ${{10}^{\left( {{10}^{10}} \right)}}$ is ${{10}^{{{10}^{8}}}}$.
Hence option (b) is correct.
Note: Alternatively we can find out the answer by cross checking the options. Take the options one by one and see if that option to the power 100 is ${{10}^{\left( {{10}^{10}} \right)}}$ or not.
Like for option (b),
${{\left( {{10}^{{{10}^{8}}}} \right)}^{100}}={{10}^{{{10}^{8}}\times 100}}={{10}^{{{10}^{8}}\times {{10}^{2}}}}={{10}^{{{10}^{8+2}}}}={{10}^{{{10}^{10}}}}$
Hence option (b) is correct.
There is a difference between the 100th root of a number and that number to the power 100.
100th root of a number, say $x$, means ${{x}^{\dfrac{1}{100}}}$.
And $x$ to the power 100 is ${{x}^{100}}$.