pre cal 11
mixed radicals to entire radical
-a√b =-√(a^2)(b) left out the negative why????
and for cube root you include the -, why is this the case?
pls help
In high school, often teachers neglect other than primary roots. Later on in engineering, they have real meaning, and the solutions have meaning. For instance, in electronic engineering cubroot(-8i) has three roots, and all three have meaning.
Here is a magic trick you will learn (out of high school). In the complex plain, 8i is 8@90, and -8i is 8(270)
so the cubrt (-8i)=cubrt(8(270))=2@90; 2@210;2@330 and all those can be converted to the complex plane (0+2i;-1.75- .935i; and one other. The point is all the roots exist, it is just a convenience to ignore some. Sometimes one gets burned doing that. We have a nice theorum that any degree n equation (cubrts are degree3, sqrts are degree2), there are n solutions. Keep that in your back pocket for later life as an engineer or scientist.
It's important to remember that √x = |x|
√9 = 3, NOT ±3
Just because (-3)^2 = 9 does not make -3 = √9
For cube roots, you can take the - sign inside or not, since (for all odd powers)
∛(-x)^3 = ∛[(-1)^3*x^3) = ∛(-1)^3 * ∛x^3 = -1*∛x^3 = -x
But √x^2 ≠ x if x < 0
@Steve So -2√4 = -√16? negative on the outside?
When simplifying mixed radicals to entire radicals, it is important to understand the properties of radicals.
For the expression -a√b, where a is a positive number and b is a positive radicand, we have a square root with a negative sign in front. By convention, we separate out the negative sign as -√b, and then simplify it as -√(a^2)(b).
Why do we separate out the negative sign? The convention is to have the negative sign outside the radical because square roots are typically defined to give only positive values. By separating the negative sign, we can avoid extracting an imaginary number from the square root.
Now, let's consider cube roots. When dealing with cube roots, we include the "-" for negative numbers because cube roots have the ability to return both positive and negative values.
For example, if we have -x^3 = -∛(x^3), it is valid to write it as -∛(-x^3) because the cube root of a negative number yields a negative result.
So, when simplifying mixed radicals to entire radicals, it is important to follow the convention for square roots to avoid extracting an imaginary number, and for cube roots, both positive and negative values are considered.