Suppose you have the expression Xa/b. Xa/b is equal to bâˆš(Xa) that is, the bth root of (X to the power a). Equivalently, it is (bâˆšX)a, that is (the bth root of X), raised to the power a.
x to the power a divided by x to the power b = x to the power (a - b), ie xa/xb = xa-b. When a = b, xa/xb = 1 and a - b = 0 so xa-b = x0. Rearranging gives x0 = 1. This is true for ALL non-zero values of x.
If A = (xa, ya) and B = (xb, yb) and xa is not equal to xb, then gradient of AB = (ya - yb)/(xb - xb).If xa = xb then the gradient is undefined.
Any number to the power zero is equal to one. That can be derived from the following index law: xa*xb = xa+b (x not zero) Now let b = 0 so that the above becomes xa*x0 = xa+0 so xa*x0 = xa (since a+0 = a) That is, any number multiplied by x0 is the number itself. That can be true only if x0 is the multiplicative identity, that is, only if x0 = 1.
It is a consequence of the definition of the index laws. xa * xb = xa+b If you put b = 0 in the above equation, then you get xa * x0 = xa+0 But a+0 = a so that the right hand side becomes xa Thus the equation now reads xa * x0 = xa For that to be true for all x, x0 must be the identity element for multiplication. That is x0 = 1 for all x.
b + a + b - a = 2b
This derives from one of the laws of indices which states that, for any x (not = 0), xa * xb = xa+b Put b = 0 Then xa * x0 = xa+0 = xa (because a + 0 = a) But that means that x0 is the multiplicative identity. And since that is unique, and equal to 1, x0 = 1. This is true for all x. Put
The exponent rule for multiplication is xa * xb = xa+b Now, if you put b = 0, then a+b = a so that the above reads: xa * x0 = xa which only works if x0 = 1.
The question is ambiguous: xa+1 * xa+1 = x2(a+1) or (xa + 1)(xa + 1) = x2a + 2xa + 1
Any number raised to the power 0 is 1. This follow from the law of multiplications of power: xa * xb = xa+b Now, if you put b = 0, you get xa + x0 = xa+0 and since a+0 = a, the right hand side is xa. So you have xa * x0 = xa and using the property of the multiplicative identity, xa = 1.