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 \(\displaystyle\sqrt{{{18}{a}}}\div\sqrt{{{2}{a}^{4}}}.\)

Answers (1)

2020-11-15
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 \(\displaystyle\frac{{{3}\sqrt{{a}}}}{{a}^{2}}.\)
0

Relevant Questions

asked 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 \(\displaystyle\sqrt{{{81}{a}^{12}{b}^{20}}}\)
asked 2021-02-13
We need to calculate: The simplified value of the radical expression \(\sqrt[3]{\sqrt[3]{2a}}\)
asked 2021-01-22
We need to calculate the simplified form of the expression,
\(\frac{c^{2}\ +\ 13c\ +\ 18}{c^{2}\ -\ 9}\ +\ \frac{c\ +\ 1}{c\ +\ 3}\ -\ \frac{c\ +\ 8}{c\ -\ 3}\)
asked 2020-12-16
To calculate: The simplified form of the expression \(\displaystyle\frac{\sqrt{{12}}}{\sqrt{{{x}+{1}}}}.\)
asked 2021-02-25
To determine:
The simplified value of the radical expression \(\displaystyle\sqrt{{{3}}}{\left\lbrace{2}{a}\right\rbrace}\cdot\sqrt{{{2}{a}}}\)
asked 2021-01-05
The simplified form of the expression
\(\displaystyle{\sqrt[{{4}}]{{{c}{d}^{2}}}}\times{\sqrt[{{3}}]{{{c}^{2}{d}}}}.\)
asked 2020-11-05
Express the radical as power.
\(\displaystyle{\left({a}\right)}{\sqrt[{{6}}]{{{x}^{5}}}}\)
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.
asked 2020-11-23
To multiply:
The given expression. Then simplify if possible. Assume that all variables represent positive real numbers.
Given:
An expression: \(\displaystyle\sqrt{{3}}{\left(\sqrt{{27}}-\sqrt{{3}}\right)}\)
asked 2020-11-08
To simplify:
The given redical:
An expression: \(\displaystyle{\sqrt[{{4}}]{{{\left({x}^{2}-{4}\right)}^{4}}}}.\)
Use absolute value bars when necessary.
asked 2021-01-22
Use rational exponents to write a single radical expression.
\(\displaystyle{\sqrt[{{7}}]{{11}}}\times{\sqrt[{{6}}]{{13}}}\)
...