SAT Mathematics Practice Test 2

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