Since the numerator and denominator of
have a common base, it follows by the laws of exponents that this expression can be rewritten as
xa2−b2.Thus,the equation
=16 can be written as
xa2−b2=x16 Because the equivalent expressions have the common base x, and
x>1 it follows that the exponents of the two expressions must also be equivalent.Hence, the equation
a2−b2=16 must be true. The left-hand side of this new equation is a difference of squares, and so it can be factored:
(a+b)(a−b)=16.Finally,dividing both sides of
2(a−b)=16 by 2 gives
a−b=8Choices B, C, and D are incorrect and may result from errors in applying the laws of exponents or errors in solving the equation
a2−b2=16.