implify each expression below using the properties of integer exponents. Assume \( p \neq 0 \) a. Verify by numerical substitution. \( \begin{array}{ll}\text { a. } \frac{x^{6}}{x^{2}} & \text { b. } \frac{p^{6}}{p^{-4}}\end{array} \)

To simplify the given expressions using the properties of integer exponents, let's break it down: For part a, we have \( \frac{x^{6}}{x^{2}} \). According to the quotient rule for exponents, when dividing like bases, you subtract the exponents. Thus, \( x^{6 - 2} = x^{4} \). For part b, we look at \( \frac{p^{6}}{p^{-4}} \). Again applying the quotient rule, we can subtract the exponents: \( p^{6 - (-4)} = p^{6 + 4} = p^{10} \). Now, let's verify these simplifications by numerical substitution. For part a, let’s substitute \( x = 2 \): \[ \frac{2^{6}}{2^{2}} = \frac{64}{4} = 16 \] And for the simplified expression \( x^{4} \): \[ 2^{4} = 16 \] Both are equal, confirming our simplification. For part b, substitute \( p = 2 \): \[ \frac{2^{6}}{2^{-4}} = \frac{64}{\frac{1}{16}} = 64 \times 16 = 1024 \] And for the simplified expression \( p^{10} \): \[ 2^{10} = 1024 \] Again both are equal, validating our simplification.