Question

an expression that contains a radical is in simplest radical form if (1) no number under a radical sign has a factor that is a perfect square, (2) no fraction is under a radical sign, and (3) no radical is in a denominator.
\frac{2sqrt{3}}{5} is in simplest radical form, because the number under the radical sign doesnt have a perfect square factor, no fraction is under the radical sign, and no radical is in the denominator.
\frac{sqrt{12}}{5} is not in simplest radical form, because the number under the radical has 4 as a factor, and 4 is a perfect square.
sqrt{\frac{3}{5}} is not in simplest radical form, because a fraction is under the radical sign.
\frac{2}{5sqrt{3}} is not in simplest radical form, because a radical sign is in the denominator.
circle the numbers which are in simplest radical form.
\frac{3sqrt{2}}{10} \frac{sqrt{6}}{4} \frac{1}{sqrt{5}} \frac{3}{8}sqrt{7} \frac{5sqrt{3}}{sqrt{7}} \frac{2}{7sqrt{7}} \frac{sqrt{8}}{3}
use the quotient rule. then simplify the numerator and denominator. circle the result if it is in simplest radical form.
sqrt{\frac{12}{25}}=\frac{sqrt{12}}{sqrt{25}}=\frac{sqrt{4}sqrt{3}}{5}=\frac{2sqrt{3}}{5}
sqrt{\frac{8}{9}}=
sqrt{\frac{3}{4}}=
sqrt{\frac{4}{27}}=
sqrt{\frac{3}{5}}=
sqrt{\frac{45}{4}}=
sqrt{\frac{1}{2}}=
sqrt{\frac{1}{12}}=
sqrt{\frac{25}{48}}=
sqrt{\frac{24}{49}}=

Explanation:

Step1: Recall the rules of simplest radical form

A radical $\sqrt[n]{a}$ is in simplest form when: (1) The radicand $a$ has no factor that is a perfect $n$ - th power, (2) there is no fraction under the radical sign, and (3) there is no radical in the denominator.

Step2: Apply the quotient rule $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ ($b

eq0$)
For example, for $\sqrt{\frac{8}{9}}$, we rewrite it as $\frac{\sqrt{8}}{\sqrt{9}}$.

Step3: Simplify the square - roots

If the radicand has a perfect - square factor, we simplify it. For $\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}$ and $\sqrt{9} = 3$.

Step4: Rationalize the denominator if necessary

If there is a radical in the denominator, we multiply the numerator and denominator by the radical in the denominator to get rid of it. For example, for $\frac{1}{\sqrt{5}}$, we multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to get $\frac{\sqrt{5}}{5}$.

Answer:

1. For $\sqrt{\frac{8}{9}}$:

• $\sqrt{\frac{8}{9}}=\frac{\sqrt{8}}{\sqrt{9}}=\frac{\sqrt{4\times2}}{3}=\frac{2\sqrt{2}}{3}$

1. For $\sqrt{\frac{3}{4}}$:

• $\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{\sqrt{4}}=\frac{\sqrt{3}}{2}$

1. For $\sqrt{\frac{4}{27}}$:

• $\sqrt{\frac{4}{27}}=\frac{\sqrt{4}}{\sqrt{27}}=\frac{2}{\sqrt{9\times3}}=\frac{2}{3\sqrt{3}}=\frac{2\sqrt{3}}{9}$

1. For $\sqrt{\frac{3}{5}}$:

• It is already in simplest radical - form as there is no perfect - square factor in 3 and no radical in the denominator and no fraction under the radical sign.

1. For $\sqrt{\frac{45}{4}}$:

• $\sqrt{\frac{45}{4}}=\frac{\sqrt{45}}{\sqrt{4}}=\frac{\sqrt{9\times5}}{2}=\frac{3\sqrt{5}}{2}$

1. For $\sqrt{\frac{1}{2}}$:

• It is already in simplest radical - form as there is no perfect - square factor in 1 and no radical in the denominator and no fraction under the radical sign.

1. For $\sqrt{\frac{1}{12}}$:

• $\sqrt{\frac{1}{12}}=\frac{\sqrt{1}}{\sqrt{12}}=\frac{1}{\sqrt{4\times3}}=\frac{1}{2\sqrt{3}}=\frac{\sqrt{3}}{6}$

1. For $\sqrt{\frac{25}{48}}$:

• $\sqrt{\frac{25}{48}}=\frac{\sqrt{25}}{\sqrt{48}}=\frac{5}{\sqrt{16\times3}}=\frac{5}{4\sqrt{3}}=\frac{5\sqrt{3}}{12}$

1. Numbers in simplest radical form among $\frac{3\sqrt{2}}{10},\frac{\sqrt{6}}{4},\frac{1}{\sqrt{5}},\frac{3}{8}\sqrt{7},\frac{5\sqrt{3}}{\sqrt{7}},\frac{2}{7\sqrt{7}},\frac{\sqrt{8}}{3}$:

• $\frac{3\sqrt{2}}{10}$: No perfect - square factor in 2, no fraction under the radical sign, no radical in the denominator.
• $\frac{\sqrt{6}}{4}$: No perfect - square factor in 6, no fraction under the radical sign, no radical in the denominator.
• $\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}$ (rationalize the denominator), originally not in simplest form.
• $\frac{3}{8}\sqrt{7}$: No perfect - square factor in 7, no fraction under the radical sign, no radical in the denominator.
• $\frac{5\sqrt{3}}{\sqrt{7}}=\frac{5\sqrt{21}}{7}$ (rationalize the denominator), originally not in simplest form.
• $\frac{2}{7\sqrt{7}}=\frac{2\sqrt{7}}{49}$ (rationalize the denominator), originally not in simplest form.
• $\frac{\sqrt{8}}{3}=\frac{2\sqrt{2}}{3}$ (simplify $\sqrt{8}$), originally not in simplest form.
• The numbers in simplest radical form are $\frac{3\sqrt{2}}{10},\frac{\sqrt{6}}{4},\frac{3}{8}\sqrt{7}$.