\( a \quad ( \frac { 16 } { 9 } ) ^ { - \frac { 1 } { 2 } } \)

To simplify the expression \( a \quad ( \frac { 16 } { 9 } ) ^ { - \frac { 1 } { 2 } } \), we can first rewrite it using properties of exponents. The negative exponent indicates that we should take the reciprocal of the base, and the \( \frac{1}{2} \) exponent suggests that we will be taking the square root. Thus, \( ( \frac { 16 } { 9 } ) ^ { - \frac { 1 } { 2 } } = \frac{1}{\sqrt{\frac{16}{9}}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \). Therefore, \( a = \frac{3}{4} \). As we dive into the history of exponents, did you know that the notation we use today was popularized in the 17th century by mathematician René Descartes? Before this, mathematicians had various ways of expressing powers, leading to a patchwork of notations. Descartes’ systematic approach made calculations easier and became the foundation we continue to use in mathematics. Now, when applying exponent laws, a common mistake is forgetting how to handle negative exponents and roots. It’s tempting to overlook the reciprocal nature of negative powers, or to confuse the operations involved in square roots. Always remember: a negative exponent means flip and simplify, and that square roots and fractions can sometimes trip you up!