#### Answer

$a=16$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the value of $
a
$ such that the given equation, $
am^2+8m+1=0
,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable.
$\bf{\text{Solution Details:}}$
In the equation above, $a=
a
,$ $b=
8
,$ and $c=
1
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is
\begin{array}{l}\require{cancel}
(8)^2-4(a)(1)
\\\\=
64-4a
.\end{array}
Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then
\begin{array}{l}\require{cancel}
64-4a=0
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
-4a=-64
\\\\
a=\dfrac{-64}{-4}
\\\\
a=16
.\end{array}
Hence, the given equation has $1$ rational solution when $
a=16
.$