allhvasstH

2021-02-05

The given expression using rational exponents. Then simplify and convert back to radical notation. Assume that all variables represent positive real numbers.
Given:
The expression is $\sqrt{81{a}^{12}{b}^{20}}$

wornoutwomanC

Key points used:
- If a is real number and is a real number.
- If a is real number and n is an integer with exists.
$-{\left(a,b\right)}^{r}={a}^{r},{b}^{r}$
- Power rule ${\left({a}^{m}\right)}^{n}={a}^{mn}$
Calculation:
Consider the expression $\sqrt{81{a}^{12}{b}^{20}}$
Write the expression $\sqrt{81{a}^{12}{b}^{20}}$ using rational exponents.
And we know that $\sqrt[n]{{a}^{m}}={\left({a}^{m}\right)}^{\frac{1}{n}}={a}^{\frac{m}{n}}$
$\sqrt{81{a}^{12}{b}^{20}}={\left(81{a}^{12}{b}^{20}\right)}^{\frac{1}{2}}$
From ${\left(a,b\right)}^{r}={a}^{r},{b}^{r},{\left(81{a}^{12}{b}^{20}\right)}^{\frac{1}{2}}={81}^{\frac{1}{2}}{\left({a}^{12}\right)}^{\frac{1}{2}},{\left({b}^{20}\right)}^{\frac{1}{2}}$
${81}^{\frac{1}{2}}{\left({a}^{12}\right)}^{\frac{1}{2}},{\left({b}^{20}\right)}^{\frac{1}{2}}={\left({9}^{2}\right)}^{\frac{1}{2}}{\left({a}^{12}\right)}^{\frac{1}{2}},{\left({b}^{20}\right)}^{\frac{1}{2}}$
And from power rule as
${\left({9}^{2}\right)}^{\frac{1}{2}}{\left({a}^{12}\right)}^{\frac{1}{2}},{\left({b}^{20}\right)}^{\frac{1}{2}}=9,{a}^{6},{b}^{10}$
Conclusion:
Therefore, $\sqrt{81{a}^{12}{b}^{20}}=9{a}^{6}{b}^{10}.$

Jeffrey Jordon