# The 100th root of ${{10}^{\left( {{10}^{10}} \right)}}$ is:

a.${{10}^{{{8}^{10}}}}$

b.${{10}^{{{10}^{8}}}}$

c.${{\left( \sqrt{10} \right)}^{{{\left( \sqrt{10} \right)}^{10}}}}$

d.$10{{\left( \sqrt{10} \right)}^{\sqrt{10}}}$

Answer

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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,

$\begin{align}

& \Rightarrow x={{\left( {{10}^{\left( {{10}^{10}} \right)}} \right)}^{\dfrac{1}{100}}} \\

& \Rightarrow x={{10}^{\dfrac{{{10}^{10}}}{100}}} \\

\end{align}$

We can write 100 as 10 multiplied by 10. Therefore,

$\begin{align}

& \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}}}} \\

\end{align}$

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 ${{x}^{100}}$.

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,

$\begin{align}

& \Rightarrow x={{\left( {{10}^{\left( {{10}^{10}} \right)}} \right)}^{\dfrac{1}{100}}} \\

& \Rightarrow x={{10}^{\dfrac{{{10}^{10}}}{100}}} \\

\end{align}$

We can write 100 as 10 multiplied by 10. Therefore,

$\begin{align}

& \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}}}} \\

\end{align}$

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 ${{x}^{100}}$.

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