"To calculate: The simplified value of the radical..." solved and explained

Dolly Robinson

Answered question

2020-11-14

To calculate: The simplified value of the radical expression

\sqrt{18 a} \div \sqrt{2 a^{4}} .

unett

Skilled2020-11-15Added 119 answers

Given expression is

\sqrt{18 a} \div \sqrt{2 a^{4}} .

Formula used:
The product rule for radicals,

\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 / n} .

Here, a is the radicand, and n is the index of the radical.
Calculation:
Consider the given expression,

\sqrt{18 a} \div \sqrt{2 a^{4}} .

Use the product rule for radicals to simplify the expression,

\sqrt{18 a} \div \sqrt{2 a^{4}} = \frac{\sqrt{18 a}}{\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{18 a} \div \sqrt{2 a^{4}} = \frac{{(3^{2})}^{1 / 2} \sqrt{a}}{{(a^{2})}^{1 / 2} {(a^{2})}^{1 / 2}}

= \frac{{(3)}^{2 (1 / 2)} \sqrt{a}}{{(a)}^{2 (1 / 2)} {(a)}^{2 (1 / 2)}}

= \frac{(3) \sqrt{a}}{(a) (a)}

= \frac{3 \sqrt{a}}{n^{2}}

Hence, the simplified value of the provided expression is

\frac{3 \sqrt{a}}{a^{2}} .

Jeffrey Jordon

Expert2021-10-25Added 2605 answers

Answer is given below (on video)

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