Question

How are exponents and radicals used to present roots of real numbers

Answer 1

`\sqrt[n]{a} =a^{(1)/(n)}`

Step-by-step explanation:

Roots of real numbers can be represented by radicals or by exponents.

First, I present some examples to show how exponents and radicals are related, and then generalize.

`√(4)=4^{((1)/(2))}=(2^2)^(1)/(2)=(2)^{(2)/(2)}=2^1=2\\\\\\√(25)=(25)^{(1)/(2)}=(5^2)^{(1)/(2)}=(5)^{(2)/(2)}=5^1=5`

`\sqrt[3]{8}=(8)^{(1)/(3)}=(2^3)^{(1)/(3)}=(2)^{(3)/(3)}=2^1=2`

When you write 5² = 25, then 5 is the square root of 25.

And in general, if n is a positive integer and
`a^n=x` , then
`a` is the nth root of x.

Also, if n even (and positive) and
`a` is positive, then
`a^{(1)/(n)}` is the positive nth root of
`a`

Thus,

`\sqrt[n]{a} =a^{(1)/(n)}`