Step 1

Let us find the expression for given blank as shown below:

\(\displaystyle\underline{{\ }}\cdot{x}^{{{\frac{{{1}}}{{{8}}}}}}={x}^{{{\frac{{{4}}}{{{8}}}}}}\) (Given expression)

\(\displaystyle\underline{{\ }}\cdot{\frac{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}={\frac{{{x}^{{{\frac{{{4}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}\) (Divide both sides by \(\displaystyle{x}^{{{\frac{{{1}}}{{{8}}}}}}\)

\(\displaystyle\Rightarrow\underline{{\ }}={\frac{{{x}^{{{\frac{{{4}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}\)

\(\displaystyle\Rightarrow\underline{{\ }}={x}^{{{\frac{{{4}}}{{{8}}}}-{\frac{{{1}}}{{{8}}}}}}\) (Applying rule \(\displaystyle{\frac{{{a}^{{{m}}}}}{{{a}^{{{n}}}}}}={a}^{{{m}-{n}}}\))

\(\displaystyle\Rightarrow\underline{{\ }}={x}^{{{\frac{{{3}}}{{{8}}}}}}\)

Therefore, the expression for the given blank would be \(\displaystyle{x}^{{{\frac{{{3}}}{{{8}}}}}}\)

Let us find the expression for given blank as shown below:

\(\displaystyle\underline{{\ }}\cdot{x}^{{{\frac{{{1}}}{{{8}}}}}}={x}^{{{\frac{{{4}}}{{{8}}}}}}\) (Given expression)

\(\displaystyle\underline{{\ }}\cdot{\frac{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}={\frac{{{x}^{{{\frac{{{4}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}\) (Divide both sides by \(\displaystyle{x}^{{{\frac{{{1}}}{{{8}}}}}}\)

\(\displaystyle\Rightarrow\underline{{\ }}={\frac{{{x}^{{{\frac{{{4}}}{{{8}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{8}}}}}}}}}\)

\(\displaystyle\Rightarrow\underline{{\ }}={x}^{{{\frac{{{4}}}{{{8}}}}-{\frac{{{1}}}{{{8}}}}}}\) (Applying rule \(\displaystyle{\frac{{{a}^{{{m}}}}}{{{a}^{{{n}}}}}}={a}^{{{m}-{n}}}\))

\(\displaystyle\Rightarrow\underline{{\ }}={x}^{{{\frac{{{3}}}{{{8}}}}}}\)

Therefore, the expression for the given blank would be \(\displaystyle{x}^{{{\frac{{{3}}}{{{8}}}}}}\)