Question

Simplify:

(1) ${18}^{\frac{1}{4}}\times {18}^{\frac{1}{5}}$
(2) $7^{\frac{1}{3}}÷7^{\frac{1}{5}}$
(3) ${\left(4^{\frac{1}{2}}\right)}^{\frac{1}{9}}$
(4) ${\left(9\times 5\right)}^{\frac{1}{6}}$
(5) $x^{\frac{1}{7}}÷x^{\frac{1}{10}}$
(6) ${\left(xy\right)}^{\frac{1}{2}}$
(7) ${\left(y^{\frac{1}{3}}\right)}^{\frac{1}{2}}$
(8) ${\left(m^{\frac{1}{3}}\right)}^{\frac{1}{7}}$
(9) $p^{\frac{1}{2}}\times p^{\frac{1}{2}}$
(10) $m^{\frac{1}{3}}\times m^{\frac{1}{3}}$
(11) ${\left(7\times 9\right)}^{\frac{1}{5}}$
(12) ${\left(mn\right)}^{\frac{1}{p}}$
(13) $x^{\frac{1}{8}}÷x^{\frac{1}{11}}$
(14) $x^{\frac{1}{2}}÷x^{\frac{1}{2}}$
(15) ${\left(13÷15\right)}^{\frac{1}{7}}$

Answer 1

(1) ${18}^{\frac{1}{4}}\times {18}^{\frac{1}{5}}$ can be expressed as: ${18}^{\frac{1}{4}}\times {18}^{\frac{1}{5}}={18}^{\left(\frac{1}{4}+\frac{1}{5}\right)}={18}^{\frac{9}{20}}$
So, ${18}^{\frac{1}{4}}\times {18}^{\frac{1}{5}}={18}^{\frac{9}{20}}$

(2) $7^{\frac{1}{3}}÷7^{\frac{1}{5}}$ can be expressed as: $7^{\frac{1}{3}}÷7^{\frac{1}{5}}=7^{\left(\frac{1}{3}-\frac{1}{5}\right)}=7^{\frac{2}{15}}$
So, $7^{\frac{1}{3}}÷7^{\frac{1}{5}}=7^{\frac{2}{15}}$

(3) ${\left(4^{\frac{1}{2}}\right)}^{\frac{1}{9}}$ can be expressed as: $4^{\frac{1}{2}\times \frac{1}{9}}=4^{\frac{1}{18}}$
So, ${\left(4^{\frac{1}{2}}\right)}^{\frac{1}{9}}=4^{\frac{1}{18}}$

(4) ${\left(9\times 5\right)}^{\frac{1}{6}}$ can be expressed as: $9^{\frac{1}{6}}\times 5^{\frac{1}{6}}$
$\therefore {\left(9\times 5\right)}^{\frac{1}{6}}=9^{\frac{1}{6}}\times 5^{\frac{1}{6}}$

(5) $x^{\frac{1}{7}}÷x^{\frac{1}{10}}$ can be expressed as: $x^{\frac{1}{7}}÷x^{\frac{1}{10}}=x^{\left(\frac{1}{7}-\frac{1}{10}\right)}=x^{\frac{3}{70}}$
So, $x^{\frac{1}{7}}÷x^{\frac{1}{10}}=x^{\frac{3}{70}}$

(6) ${\left(xy\right)}^{\frac{1}{2}}$ can be expressed as: ${\left(x\times y\right)}^{\frac{1}{2}}=x^{\frac{1}{2}}\times y^{\frac{1}{2}}$
$\therefore {\left(xy\right)}^{\frac{1}{2}}=x^{\frac{1}{2}}{y}^{\frac{1}{2}}$

(7) ${\left(y^{\frac{1}{3}}\right)}^{\frac{1}{2}}$ can be expressed as: $y^{\frac{1}{3}\times \frac{1}{2}}=y^{\frac{1}{6}}$
So, ${\left(y^{\frac{1}{3}}\right)}^{\frac{1}{2}}=y^{\frac{1}{6}}$

(8) ${\left(m^{\frac{1}{3}}\right)}^{\frac{1}{7}}$ can be expressed as: $m^{\frac{1}{3}\times \frac{1}{7}}=m^{\frac{1}{21}}$
So, ${\left(m^{\frac{1}{3}}\right)}^{\frac{1}{7}}=m^{\frac{1}{21}}$

(9) $p^{\frac{1}{2}}\times p^{\frac{1}{2}}$ can be expressed as: $p^{\frac{1}{2}+\frac{1}{2}}=p^1$
$\therefore p^{\frac{1}{2}}\times p^{\frac{1}{2}}=p$

(10) $m^{\frac{1}{3}}\times m^{\frac{1}{3}}$ can be expressed as: $m^{\frac{1}{3}+\frac{1}{3}}=m^{\frac{2}{3}}$
$\therefore m^{\frac{1}{3}}\times m^{\frac{1}{3}}=m^{\frac{2}{3}}$

(11) ${\left(7\times 9\right)}^{\frac{1}{5}}$ can be expressed as: $7^{\frac{1}{5}}\times 9^{\frac{1}{5}}$
$\therefore {\left(7\times 9\right)}^{\frac{1}{5}}=7^{\frac{1}{5}}\times 9^{\frac{1}{5}}$

(12) ${\left(mn\right)}^{\frac{1}{p}}$ can be expressed as: ${\left(m\times n\right)}^{\frac{1}{p}}=m^{\frac{1}{p}}\times n^{\frac{1}{p}}$
$\therefore {\left(mn\right)}^{\frac{1}{p}}=m^{\frac{1}{p}}{n}^{\frac{1}{p}}$

(13) $x^{\frac{1}{8}}÷x^{\frac{1}{11}}$ can be expressed as: $x^{\frac{1}{8}}÷x^{\frac{1}{11}}=x^{\left(\frac{1}{8}-\frac{1}{11}\right)}=x^{\frac{3}{88}}$
So, $x^{\frac{1}{8}}÷x^{\frac{1}{11}}=x^{\frac{3}{88}}$

(14) $x^{\frac{1}{2}}÷x^{\frac{1}{2}}$ can be expressed as: $x^{\frac{1}{2}}÷x^{\frac{1}{2}}=x^{\left(\frac{1}{2}-\frac{1}{2}\right)}=x^0=1$
So, $x^{\frac{1}{2}}÷x^{\frac{1}{2}}=1$

(15) ${\left(13÷15\right)}^{\frac{1}{7}}$ can be expressed as: ${13}^{\frac{1}{7}}÷{15}^{\frac{1}{7}}$
$\therefore {\left(13÷15\right)}^{\frac{1}{7}}={13}^{\frac{1}{7}}÷{15}^{\frac{1}{7}}$