Answer
$\dfrac{\sqrt[4]{7}+t^2\sqrt[]{3}}{t^{3}} $
Work Step by Step
$\bf{\text{Solution Outline:}}$
Simplify each term in the given expression, $ \sqrt[4]{\dfrac{7}{t^{12}}}+\sqrt[4]{\dfrac{9}{t^4}} .$ Then make the terms similar (same denominator) to combine the numerators.
$\bf{\text{Solution Details:}}$
Simplifying each term of the expression above results to \begin{array}{l}\require{cancel} \sqrt[4]{\dfrac{1}{t^{12}}\cdot7}+\sqrt[4]{\dfrac{1}{t^4}\cdot9} \\\\= \sqrt[4]{\left(\dfrac{1}{t^{3}}\right)^4\cdot7}+\sqrt[4]{\left(\dfrac{1}{t}\right)^4\cdot9} \\\\= \dfrac{1}{t^{3}}\sqrt[4]{7}+\dfrac{1}{t}\sqrt[4]{9} \\\\= \dfrac{\sqrt[4]{7}}{t^{3}}+\dfrac{\sqrt[4]{9}}{t} .\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression $\sqrt[4]{9}$ above is equivalent to
\begin{array}{l}\require{cancel}
9^{\frac{1}{4}}
\\\\=
(3^2)^{\frac{1}{4}}
\\\\=
3^{2\cdot\frac{1}{4}}
\\\\=
3^{\frac{2}{4}}
\\\\=
3^{\frac{1}{2}}
\\\\=
\sqrt{3}
.\end{array}
Hence, the expression, $\dfrac{\sqrt[4]{7}}{t^{3}}+\dfrac{\sqrt[4]{9}}{t},$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[4]{7}}{t^{3}}+\dfrac{\sqrt[]{3}}{t}
.\end{array}
To simplify the expression above, make the terms similar by multiplying the necessary term/s to an expression equal to $1$. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[4]{7}}{t^{3}}+\dfrac{\sqrt[]{3}}{t}\cdot\dfrac{t^2}{t^2} \\\\= \dfrac{\sqrt[4]{7}}{t^{3}}+\dfrac{t^2\sqrt[]{3}}{t^3} .\end{array} To combine similar terms, add/subtract the numerators and copy the similar denominator. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[4]{7}+t^2\sqrt[]{3}}{t^{3}} .\end{array}
