Just Subtract By One

Algebra Level 2

What is the $5^\text{th}$ root of $\large 5^{5^{5^5}}$ ?

5^{5^{5^4}}

5^{5^{4^5}}

5^{4^{5^5}}

4^{5^{5^5}}

None of these choices

1 solution

Nihar Mahajan
Apr 8, 2016

$\huge\sqrt[5]{5^{5^{5^{5}}}}=5^{\frac{5^{5^5}}{5}} = 5^{5^{5^5-1}} \longrightarrow \text{None of These}$

Moderator note:

Good approach of explaining how to deal with this tower of exponents.

For further explanation: we use the two rules of exponents : $\sqrt[m]{a^n} = a^{n/m}$ and $a^{m-n} = a^m \div a^n$ .

So the $5^\text{th}$ root of $a = 5^{5^{5^5}}$ is equal to $\LARGE a^{1/5} = 5^{5^{5^5} \div 5} = 5^{5^{5^5} \div 5^1} = 5^{5^{5^5-1} } \; ,$ or equivalently, $\LARGE 5^{5^{\left( -1+5^5 \right)}}$ .

Relevant wiki: How are exponent towers evaluated?

Pi Han Goh - 5 years, 2 months ago

we will not multiply 1/5 with one of the 5 (in the power) .

For example:

$((2^{2})^{2}) ^{1/}{2}$ = $(2^{4})^{1/}{2}= \boxed{2^{2}}$ $\huge{But}$ $2^{2^{2^{2}}} = 2^{8}$

$\therefore \huge{ \sqrt[5]{5^{5^{5^{5}}}} = ({5^{5^{5^{5}}}})^{1/5} }$ = first example

I am not good at this :(

Syed Baqir - 5 years, 2 months ago

Just Subtract By One

1 solution

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