Today's soundtrack is Lil Wayne: Tha Carter V, an album that shows how much better the father of mumble rap is at rapping than he is at playing guitar.
To arrange radicals in order from least to greatest, if they have coefficients, first we put the same number of coefficients inside of the radical as the index value - for example, 5√25 = √5x5x25.
We can express radicals in different forms: either as mixed radicals or entire radicals. We must remember that a fraction radicand is treated the same as if both the numerator and the denominator are each a radicand.
Sometimes we will be asked to determine how a radical is defined if it includes a variable radicand. What this is asking is, "What properties must the variable have for this radical to work, or be 'real'?" There are three ways that we can define radicals:
- x ∈ ℝ: the variable is a real number
- - Use with even index and even exponent
- - Use with odd index
- x ≥ 0: the variable is greater than or equal to zero
- - Use with even index and odd exponent
- x ≤ 0: the variable is less than or equal to zero
- - Use with even index and odd exponent if radicand is negative
When we solve a variable within a radical, if both the radical's index and the radicand's exponent are even, we show that the solution is absolute thus: |x|, which is the same as saying that it could be either positive or negative and it would still work (±x).