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Communicate:
#20 Explain in full sentences why the radical expression is or is not in simplifie form. Give more than one reason to back up your answer.

(sqrt(12 n))/(sqrt(32 n))

Communicate: $\newline$ \# $20$ Explain in full sentences why the radical expression is or is not in simplifie form. Give more than one reason to back up your answer. $\newline$ $\frac{\sqrt{12 n}}{\sqrt{32 n}}$

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Simplify radical expressions with root inside the root

Full solution

Q. Communicate:

\newline

\#

20

Explain in full sentences why the radical expression is or is not in simplifie form. Give more than one reason to back up your answer.

\newline

\frac{\sqrt{12 n}}{\sqrt{32 n}}

Factor and Simplify: First, let's factor the numbers under the radicals to see if they can be simplified. $\newline$ $\sqrt{12 n} = \sqrt{4 \times 3 \times n} = \sqrt{4} \times \sqrt{3n}$ $\newline$ $\sqrt{32 n} = \sqrt{16 \times 2 \times n} = \sqrt{16} \times \sqrt{2n}$
Simplify Perfect Squares: Now, simplify the square roots of the perfect squares. $\sqrt{4} = 2$ and $\sqrt{16} = 4$ So, $\frac{\sqrt{12 n}}{\sqrt{32 n}}$ becomes $\frac{2 \cdot \sqrt{3n}}{4 \cdot \sqrt{2n}}$
Simplify Fraction: Next, we can simplify the fraction by dividing both the numerator and the denominator by $2$ . $(2 \times \sqrt{3n}) / (4 \times \sqrt{2n}) = (\sqrt{3n}) / (2 \times \sqrt{2n})$
Divide Radicands: Now, let's simplify the radical expression further by dividing the radicands (the numbers under the radicals). $\frac{\sqrt{3n}}{\sqrt{2n}} = \sqrt{\frac{3n}{2n}}$
Simplify Expression: Simplify the expression under the radical. $\sqrt{\frac{3n}{2n}} = \sqrt{\frac{3}{2}}$
Check for Simplification: Finally, we check if the expression $\sqrt{\frac{3}{2}}$ is in its simplest form. $\newline$ Since $3$ and $2$ are both prime numbers and there are no perfect square factors, the expression cannot be simplified further.

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