Question

# To calculate: \frac{(9a^{3}b^{4})^{\frac{1}{2}}}{15a^{2}b}

To calculate: $$\frac{(9a^{3}b^{4})^{\frac{1}{2}}}{15a^{2}b}$$

2021-08-08
Step 1
This algebraic expression needs to be simplified.
$$\displaystyle{\frac{{{\left({9}{a}^{{{3}}}{b}^{{{4}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{15}{a}^{{{2}}}{b}}}}$$
Step 2
Our first step would be to distribute the exponent 1/2 on all the terms in the numerator as shown below:
$$\displaystyle{\frac{{{\left({9}{a}^{{{3}}}{b}^{{{4}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{15}{a}^{{{2}}}{b}}}}\Rightarrow{\frac{{{\left({9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}{\left({a}^{{{3}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}{\left({b}^{{{4}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{15}{a}^{{{2}}}{b}}}}$$
Step 3
Our next step is to simplify the exponents in the numerator as shown below:
$$\displaystyle{\frac{{{\left({9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}{\left({a}^{{{3}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}{\left({b}^{{{4}}}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{15}{a}^{{{2}}}{b}}}}\Rightarrow{\frac{{{3}{a}^{{{\frac{{{3}}}{{{2}}}}}}{b}^{{{2}}}}}{{{15}{a}^{{{2}}}{b}}}}$$
Step 4
Our next step is to split the corresponding terms and write them as numerator/denominator and finally use quotient rule of exponents to simplify them as shown below:
$$\displaystyle{\frac{{{3}{a}^{{{\frac{{{3}}}{{{2}}}}}}{b}^{{{2}}}}}{{{15}{a}^{{{2}}}{b}}}}\Rightarrow{\frac{{{3}}}{{{15}}}}\cdot{\frac{{{a}^{{{\frac{{{3}}}{{{2}}}}}}}}{{{a}^{{{2}}}}}}\cdot{\frac{{{b}^{{{2}}}}}{{{b}}}}$$
$$\displaystyle\Rightarrow{\frac{{{1}}}{{{5}}}}\cdot\ {a}^{{{\frac{{{3}}}{{{2}}}}-{2}}}\cdot\ {b}^{{{2}-{1}}}\Rightarrow{\frac{{{1}}}{{{5}}}}\cdot\ {a}^{{-{\frac{{{1}}}{{{2}}}}}}\cdot\ {b}^{{{1}}}$$
Step 5
We don't want to keep the negative exponents in our answer so we shift the term with negative exponent in the denominator as shown below:
$$\displaystyle\Rightarrow{\frac{{{1}}}{{{5}}}}{a}^{{-{\frac{{{1}}}{{{2}}}}}}{b}^{{{1}}}\Rightarrow{\frac{{{b}}}{{{5}{a}^{{{\frac{{{1}}}{{{2}}}}}}}}}$$
Step 6
Finally, we don't want to keep fractional exponents (radicals) in the denominator, so we first rewrite the the exponent $$\displaystyle\frac{{1}}{{2}}$$ as a squareroot and then rationalize it as shown below:
$$\displaystyle{\frac{{{b}}}{{{5}{a}^{{{\frac{{{1}}}{{{2}}}}}}}}}\Rightarrow{\frac{{{b}}}{{{5}\sqrt{{{a}}}}}}$$
$$\displaystyle\Rightarrow{\frac{{{b}}}{{{5}\sqrt{{{a}}}}}}\cdot{\frac{{\sqrt{{{a}}}}}{{\sqrt{{{a}}}}}}$$ (Upon rationalizing)
$$\displaystyle\Rightarrow{\frac{{{b}\sqrt{{{a}}}}}{{{5}{a}}}}$$
Step 7
Answer: $$\displaystyle{\frac{{{b}\sqrt{{{a}}}}}{{{5}{a}}}}$$