Answer
$6+18a$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
