# To fill: The correct expression in the box for each equation An eq

To fill: The correct expression in the box for each equation
An equation: $$\displaystyle\underline{{\ }}\cdot{x}^{{{\frac{{{1}}}{{{8}}}}}}={x}^{{{\frac{{{4}}}{{{8}}}}}}$$, or $$\displaystyle{x}^{{{\frac{{{1}}}{{{2}}}}}}$$

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Howell
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 $$\displaystyle{x}^{{{\frac{{{3}}}{{{8}}}}}}$$