Question

Find the indicated real n roots of a N=4, a=81

Answer 1

$81|3\\27|3\\.\ 9|3\\.\ 3|3\\.\ 1|\\\\81=\underbrace{3\cdot3\cdot3\cdot3}_{4}=3^4\\\\\sqrt[4]{81}=\sqrt[4]{3^4}=3\\\\Used:\\\sqrt[n]{a^n}=a$

Answer 2

If I understood coorectly, you're looking for the fourth root of 81. This exercise can be solved by remembering that extracting the fourth root of a number is the same as raising that number to the power of 1/4.

We also need the prime factorization of 81, which is

$81 = 9 \times 9 = 3^2 \times 3^2 = 3^4$

So, the fourth root of 81 is 81 raised to the power of 1/4, which means

$\sqrt[4]{81} = \sqrt[4]{3^4} = (3^4)^{\frac{1}{4}}$

Now, use the property of exponents $(a^b)^c = a^{bc}$ to convert the expression into

$(3^4)^{\frac{1}{4}} = 3^{4\cdot \frac{1}{4}} = 3^1 = 3$