So many 5's

The fifth root of $5^{5^5}$ is equal to $5^{5^5\times\frac{1}{5}}=\boxed{5^{5^4}}$

The expression can be written as:

= $5^{5^{5} \times \frac{1}{5}}$

= $5^{5^{5} \times 5^{-1}}$

= $5^{5^{5-1}}$

= $\boxed{5^{5^{4}}}$

Given the form of the multiple-choice answers we have, we are asked to find $p$ such that $(5^p)^5=5^{5^5}$ . (I)

Observe that $(5^p)^5=5^p\times5^p\times5^p\times5^p\times5^p$

We may add sum the exponents to find the product of powers:

$(5^p)^5=5^{(p+p+p+p+p)}$ $=5^{5p}$

We now return to (I), with a substitution on the left.

$5^{5p}=5^{5^5}$

Equating the exponents,

$5p=5^5$

$p=\frac{5^5}{5}$

$=\frac{5^5}{5^1}$

$=5^{5-1}=5^4$ , so we have as our answer $\boxed{5^{5^4}}$ .

• $5 ^ {5 ^ 5}=5 ^ {3125}$
• $\sqrt[n]{a ^ k}=a ^ {\frac{k}{n}}$
• $\sqrt[5]{5 ^ {3125}}=5 ^ {\frac{3125}{5}}=5 ^ {625}=5 ^ {5 ^ 4}$

$\space\space\space\space \sqrt[5]{5^{5^{5}}} \\ = {(5^{5^5})}^{\frac{1}{5}} \\ = 5^{5^5 \times \frac{1}{5}} \\ = 5^{5^5 \times 5^{-1}} \\ = 5^{5^{5 - 1}} \\ = 5^{5^4}$

just make it 5^3125 and then find out the fifth root !

let 5^(5^5) = a, and b = 5^5, => a^(1/5) = 5^(b * 1/5) = 5^(5^(5-1)) = 5^(5^4).

Finding a fifth root is just like dividing the exponent by 5. 5^5=5^4.

(5^(5^5))^(1/5)=(5^3125)^(1/5)=5^625=5^(5^4)