## Intermediate Algebra (12th Edition)

$6+18a$
$\bf{\text{Solution Outline:}}$ Use the Distributive Property and the laws of exponents to simplify the given expression, $6a^{7/4} \left( a^{-7/4}+3a^{-3/4} \right) .$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} 6a^{7/4} ( a^{-7/4})+6a^{7/4}(3a^{-3/4} ) .\end{array} Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to \begin{array}{l}\require{cancel} 6a^{\frac{7}{4}+\left(-\frac{7}{4}\right)}+18a^{\frac{7}{4}+\left(-\frac{3}{4}\right)} \\\\= 6a^{\frac{7}{4}-\frac{7}{4}}+18a^{\frac{7}{4}-\frac{3}{4}} \\\\= 6a^{0}+18a^{\frac{4}{4}} \\\\= 6a^{0}+18a^{1} \\\\= 6a^{0}+18a .\end{array} Since any expression (except $0$) raised to zero is $1$, the expression above is equivalent to \begin{array}{l}\require{cancel} 6(1)+18a \\\\= 6+18a .\end{array}