Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Law of division of indices

a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

${\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}$ = ${\mathrm{a}}^{-\frac{1}{4}+\frac{1}{2}}$

${\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}$ = ${\mathrm{a}}^{\frac{\mathrm{-1\; x\; 1\; +\; 2\; x\; 1}}{4}}$

${\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}$ = ${\mathrm{a}}^{\frac{\mathrm{-1\; +\; 2}}{4}}$

${\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}$ = ${\mathrm{a}}^{\frac{1}{4}}$

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = $\frac{{\mathrm{a}}^{\frac{1}{4}}}{{\mathrm{a}}^{-\frac{1}{2}}}$

Apply a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = ${\mathrm{a}}^{\frac{1}{4}-\; -\frac{1}{2}}$

Note: -- = +

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = ${\mathrm{a}}^{\frac{1}{4}+\frac{1}{2}}$

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = ${\mathrm{a}}^{\frac{\mathrm{1\; x\; 1\; +\; 2\; x\; 1}}{4}}$

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = ${\mathrm{a}}^{\frac{\mathrm{1\; +\; 2}}{4}}$

$\frac{{\mathrm{a}}^{-\frac{1}{4}}x\; {\mathrm{a}}^{\frac{1}{2}}}{{\mathrm{a}}^{-\frac{1}{2}}}$ = ${\mathrm{a}}^{\frac{3}{4}}$