Answer
$p-\dfrac{1}{p}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
(p^{1/2}-p^{-1/2})(p^{1/2}+p^{-1/2})
,$ use the special product on the sum and difference of like terms. Then, use the laws of exponents to simplify the resulting expression.
$\bf{\text{Solution Details:}}$
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent\begin{array}{l}\require{cancel}
(p^{1/2})^2-(p^{-1/2})^2
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
p^{\frac{1}{2}\cdot2}-p^{-\frac{1}{2}\cdot2}
\\\\=
p^{1}-p^{-1}
\\\\=
p-p^{-1}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
p-\dfrac{1}{p^1}
\\\\=
p-\dfrac{1}{p}
.\end{array}
