# To calculate: The simplified value of the radical expression displaystylesqrt{{{18}{a}}}divsqrt{{{2}{a}^{4}}}.

Question
To calculate: The simplified value of the radical expression $$\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}.$$

2020-11-15
Given expression is $$\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}.$$
Formula used:
$$\displaystyle{\sqrt[{{n}}]{{a}}}{b}={\sqrt[{{n}}]{{a}}}\cdot{\sqrt[{{n}}]{{b}}}$$
Here, n is the positive integer and a and b are real number.
The relation between radical and rational exponent notation is expressed as,
$$\displaystyle{\sqrt[{{n}}]{{a}}}={a}^{{{1}\text{/}{n}}}.$$
Here, a is the radicand, and n is the index of the radical.
Calculation:
Consider the given expression,
$$\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}.$$
Use the product rule for radicals to simplify the expression,
$$\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}=\frac{\sqrt{{{18}{a}}}}{\sqrt{{{2}{a}^{4}}}}$$
$$\displaystyle=\frac{{\sqrt{{{9}\cdot{2}\cdot{a}}}}}{{\sqrt{{2}}\cdot{a}^{2}\cdot{a}^{2}}}$$
$$\displaystyle=\frac{{\sqrt{{{3}^{2}}}\cdot\sqrt{{2}}\cdot\sqrt{{a}}}}{{\sqrt{{2}}\cdot\sqrt{{{a}^{2}\cdot\sqrt{{{a}^{2}}}}}}}$$
Use the relation between radical and rational exponent notation to simplify the expression,
$$\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}=\frac{{{\left({3}^{2}\right)}^{{{1}\text{/}{2}}}\sqrt{{a}}}}{{{\left({a}^{2}\right)}^{{{1}\text{/}{2}}}{\left({a}^{2}\right)}^{{{1}\text{/}{2}}}}}$$
$$\displaystyle=\frac{{{\left({3}\right)}^{{{2}{\left({1}\text{/}{2}\right)}}}\sqrt{{a}}}}{{{\left({a}\right)}^{{{2}{\left({1}\text{/}{2}\right)}}}{\left({a}\right)}^{{{2}{\left({1}\text{/}{2}\right)}}}}}$$
$$\displaystyle=\frac{{{\left({3}\right)}\sqrt{{a}}}}{{{\left({a}\right)}{\left({a}\right)}}}$$
$$\displaystyle=\frac{{{3}\sqrt{{a}}}}{{n}^{2}}$$
Hence, the simplified value of the provided expression is $$\displaystyle\frac{{{3}\sqrt{{a}}}}{{a}^{2}}.$$

### Relevant Questions

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 $$\displaystyle\sqrt{{{81}{a}^{12}{b}^{20}}}$$
We need to calculate: The simplified value of the radical expression $$\sqrt[3]{\sqrt[3]{2a}}$$
We need to calculate the simplified form of the expression,
$$\frac{c^{2}\ +\ 13c\ +\ 18}{c^{2}\ -\ 9}\ +\ \frac{c\ +\ 1}{c\ +\ 3}\ -\ \frac{c\ +\ 8}{c\ -\ 3}$$
To calculate: The simplified form of the expression $$\displaystyle\frac{\sqrt{{12}}}{\sqrt{{{x}+{1}}}}.$$
To determine:
The simplified value of the radical expression $$\displaystyle\sqrt{{{3}}}{\left\lbrace{2}{a}\right\rbrace}\cdot\sqrt{{{2}{a}}}$$
The simplified form of the expression
$$\displaystyle{\sqrt[{{4}}]{{{c}{d}^{2}}}}\times{\sqrt[{{3}}]{{{c}^{2}{d}}}}.$$
$$\displaystyle{\left({a}\right)}{\sqrt[{{6}}]{{{x}^{5}}}}$$
To simplify:
The expression $$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}$$ and express the answer using rational exponents.
(b) $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$
To simplify:
The expression $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$ and express the answer using rational exponents.
An expression: $$\displaystyle\sqrt{{3}}{\left(\sqrt{{27}}-\sqrt{{3}}\right)}$$
An expression: $$\displaystyle{\sqrt[{{4}}]{{{\left({x}^{2}-{4}\right)}^{4}}}}.$$
$$\displaystyle{\sqrt[{{7}}]{{11}}}\times{\sqrt[{{6}}]{{13}}}$$