#### Answer

$x=3$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\log_{16}\sqrt{x+1}=\dfrac{1}{4}
,$ change to exponential form. Use the definition of rational exponents to simplify the result. Then square both sides and use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since $\log_by=x$ is equivalent to $y=b^x$, the equation above, in exponential form, is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{x+1}=16^{\frac{1}{4}}
.\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 above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{x+1}=\sqrt[4]{16}
\\\\
\sqrt{x+1}=\sqrt[4]{2^4}
\\\\
\sqrt{x+1}=2
.\end{array}
Squaring both sides and using the properties of equality to isolate the variable result to
\begin{array}{l}\require{cancel}
(\sqrt{x+1})^2=2^2
\\\\
x+1=4
\\\\
x=4-1
\\\\
x=3
.\end{array}