91Ó°ÊÓ

We have given,
$\left(a\right)\sqrt[4]{r^7}\left(b\right){\left(\sqrt[5]{2pq}\right)}^3\left(c\right)\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}$

1) If $\sqrt[n]{a}$is a real number and $nâ‰\yen 2$then $\sqrt[n]{a}=a^{\frac{1}{n}}$

2) Rational exponent:

For any positive integer n and m ${\left(\sqrt[n]{a}\right)}^m=a^{\frac{m}{n}}$

3) Product to a Power:

${\left(ab\right)}^m=a^m{b}^m$

4) Power Property:

${\left(a^m\right)}^n=a^{mn}$

5) Quotient to a Power Property:

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m},b≠0$

We have given,

$\sqrt[4]{r^7}$

Using the rational exponent property,

$\sqrt[4]{r^7}=r^{\frac{7}{4}}$

Hence, rational exponent of $\sqrt[4]{r^7}$is$r^{\frac{7}{4}}$.

We have given,

${\left(\sqrt[5]{2pq}\right)}^3$

Using the rational exponent rule,

${\left(\sqrt[5]{2pq}\right)}^3={\left(2pq\right)}^{\frac{3}{5}}$

Using Product to a power rule,

${\left(\sqrt[5]{2pq}\right)}^3={2^{\frac{3}{5}}{p}^{\frac{3}{5}}q}^{\frac{3}{5}}={8^{\frac{1}{5}}{p}^{\frac{3}{5}}q}^{\frac{3}{5}}$

Hence, the rational exponent of${\left(\sqrt[5]{2pq}\right)}^3$is${8^{\frac{1}{5}}{p}^{\frac{3}{5}}q}^{\frac{3}{5}}$.

We have given,

$\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}$

Using the rational exponent rule,

$\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}={\left(\frac{12m}{7n}\right)}^{\frac{3}{4}}$

Using Quotient to a power rule,

$\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}=\frac{{\left(12m\right)}^{\frac{3}{4}}}{{\left(7n\right)}^{\frac{3}{4}}}$

Using the product to power rule,

$\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}=\frac{{{\left(12\right)}^{\frac{3}{4}}m}^{\frac{3}{4}}}{{\left(7\right)}^{\frac{3}{4}}{n}^{\frac{3}{4}}}$

Hence, the rational exponent of $\sqrt[4]{{\left(\frac{12m}{7n}\right)}^3}$is $\frac{{{\left(12\right)}^{\frac{3}{4}}m}^{\frac{3}{4}}}{{\left(7\right)}^{\frac{3}{4}}{n}^{\frac{3}{4}}}$.