 # To calculate: The simplified value of the radical expression displaystylesqrt{{{18}{a}}}divsqrt{{{2}{a}^{4}}}. Dolly Robinson 2020-11-14 Answered
To calculate: The simplified value of the radical expression $\sqrt{18a}÷\sqrt{2{a}^{4}}.$
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Given expression is $\sqrt{18a}÷\sqrt{2{a}^{4}}.$
Formula used:
$\sqrt[n]{a}b=\sqrt[n]{a}\cdot \sqrt[n]{b}$
Here, n is the positive integer and a and b are real number.
The relation between radical and rational exponent notation is expressed as,
$\sqrt[n]{a}={a}^{1\text{/}n}.$
Here, a is the radicand, and n is the index of the radical.
Calculation:
Consider the given expression,
$\sqrt{18a}÷\sqrt{2{a}^{4}}.$
Use the product rule for radicals to simplify the expression,
$\sqrt{18a}÷\sqrt{2{a}^{4}}=\frac{\sqrt{18a}}{\sqrt{2{a}^{4}}}$
$=\frac{\sqrt{9\cdot 2\cdot a}}{\sqrt{2}\cdot {a}^{2}\cdot {a}^{2}}$
$=\frac{\sqrt{{3}^{2}}\cdot \sqrt{2}\cdot \sqrt{a}}{\sqrt{2}\cdot \sqrt{{a}^{2}\cdot \sqrt{{a}^{2}}}}$
Use the relation between radical and rational exponent notation to simplify the expression,
$\sqrt{18a}÷\sqrt{2{a}^{4}}=\frac{{\left({3}^{2}\right)}^{1\text{/}2}\sqrt{a}}{{\left({a}^{2}\right)}^{1\text{/}2}{\left({a}^{2}\right)}^{1\text{/}2}}$
$=\frac{{\left(3\right)}^{2\left(1\text{/}2\right)}\sqrt{a}}{{\left(a\right)}^{2\left(1\text{/}2\right)}{\left(a\right)}^{2\left(1\text{/}2\right)}}$
$=\frac{\left(3\right)\sqrt{a}}{\left(a\right)\left(a\right)}$
$=\frac{3\sqrt{a}}{{n}^{2}}$
Hence, the simplified value of the provided expression is $\frac{3\sqrt{a}}{{a}^{2}}.$
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