Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. a) r^{^1/_6} r^{^5/_6} b) a^{^3/_5} a^{^3/_{10}}

a) $a^{\frac{m}{n}}=(\sqrt[n]{a}{)}^m=\sqrt[n]{a^m}$
The exponent ${}^m{/}_n$ consists of two parts:
The denominator n is the root and the numerator m is the export
Exponent Rule:
$a^{{-}^m{/}_n}=\frac{1}{a^{{}^m{/}_n}}$
Calculation:
$r^{\frac{1}{6}}\cdot r^{\frac{5}{6}}=r$
Apply exponent rule simplify the expression,
$r^{\frac{1}{6}}\cdot r^{\frac{5}{6}}=r^{\frac{1}{6}+\frac{5}{6}}$
$=r^{\frac{6}{6}}$
$r^{\frac{1}{6}}\cdot r^{\frac{5}{6}}=r$
Final statement (a):
$r^{\frac{1}{6}}\cdot r^{\frac{5}{6}}=r$
b) $a^{\frac{m}{n}}=(\sqrt[n]{a}{)}^m=\sqrt[n]{a^m}$
The exponent m//n consists of two parts:
The denominator n is the root and the numerator m is the export
Exponent Rule:
$a^{{-}^m{/}_n}=\frac{1}{a^{{}^m{/}_n}}$
Calculation:
$a^{{}^3{/}_5}{a}^{{}^3{/}_{10}}$
Apply exponent rule in $a^{{}^3{/}_5}{a}^{{}^3{/}_{10}}$ we get,
$a^{\frac{3}{5}}\cdot a^{\frac{3}{10}}=a^{\frac{3}{5}+\frac{3}{10}}$
$=a^{\frac{3\times 2}{5\times 2}+\frac{3}{10}}$
$a^{\frac{6}{10}+\frac{3}{10}}$
$a^{\frac{3}{5}}\cdot a^{\frac{3}{10}}=a^{\frac{9}{10}}$
Final statement (b):
$a^{\frac{3}{5}}\cdot a^{\frac{3}{10}}=a^{\frac{9}{10}}$