Question

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

Answer 1

Given expression is \(\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}.\)
Formula used:
The product rule for radicals,
\(\displaystyle{\sqrt[{{n}}]{{a}}}{b}={\sqrt[{{n}}]{{a}}}\cdot{\sqrt[{{n}}]{{b}}}\)
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 \(\displaystyle\frac{{{3}\sqrt{{a}}}}{{a}^{2}}.\)