Question 616442
If your expression is
{{{root(1976, (2a+b)^1976)}}}
then there is no square root sign. The proper name for what you call a "square root sign" is "radical". Radical symbols are used for all kinds of roots:
Square roots: {{{sqrt(3x-6)}}} or {{{root(2, 4x+4)}}}
Cube roots: {{{root(3, 10x+23)}}}
4th roots: {{{root(4, -3x+90)}}}
etc.
Your expression is a 1976th root.<br>
Since 1976 is an even number, a 1976th root is an even-numbered root. Even-numbered roots are not negative. So we must make sure that whatever answer we get, it must not turn out to be negative.<br>
So how do we go about simplifying this? Well, once you understand what roots are, then this problem is <i>extremely</i> easy to simplify. A 1976th root is whatever you have to raise to the 1976th power to get the radicand as a result. ("Radicand" is the name for the expression inside a radical.) So
{{{root(1976, (2a+b)^1976)}}}
represents whatever you raise to the 1976th power to get {{{(2a+b)^1976}}}. But can you just see what was raised to the 1976th power to get {{{(2a+b)^1976}}}?? Isn't it simply (2a+b)???<br>
So at first look, {{{root(1976, (2a+b)^1976)}}} would seem to be 2a+b.<br>
But as mentioned earlier, a 1976th root must never be negative. Could 2a+b be negative? Since we don't know what a or b are, we don't know. To ensure the "non-negativeness" of our answer we must use absolute value. So the correct answer is:
{{{root(1976, (2a+b)^1976) = abs(2a+b)}}}<br>
P.S. Odd-numbered roots can be any number: positive, negative or zero. So never use absolute value on odd-numbered roots. For example:
{{{root(5033, (3x-9)^5033) = 3x-9}}}<br>
P.P.S. You don't always need to use absolute values on even-numbered roots. If you know your answer cannot be negative then the absolute value is not needed. For example:
{{{root(40, 435^40) = 435}}} (435 is obviously positive. No absolute value needed here.)
{{{root(32, (x^2+1)^32) = x^2 + 1}}} (Although we don't know what x might be, we know that {{{x^2}}} will never be negative. And if {{{x^2}}} cannot be negative, then {{{x^2+1}}} cannot be negative either. So no absolute value is needed.)