Square root of a quotient

To solve the problem, we will apply the rules of roots, specifically the Square Root Quotient Property:

• Step 1: The given expression is $\sqrt{a}:\sqrt{b}$, which represents the division of the square roots.
• Step 2: Apply the square root quotient property: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.
• Step 3: In terms of ratio notation, $\sqrt{a}:\sqrt{b}$ simplifies to $\sqrt{a:b}$.

Therefore, the expression $\sqrt{a}:\sqrt{b}$ is equivalent to $\sqrt{a:b}$, which is represented by choice 1.

Let's use the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

$$when we remember that in a square root (also called "root to the power of 2") we don't write the root's power and:

$n=2$

meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

We will convert therefore all the roots in the problem to powers:

$\frac{\sqrt{36}}{\sqrt{9}}=\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Now let's recall the power law for a fraction in parentheses:

$\frac{a^n}{c^n}= \big(\frac{a}{c}\big)^n$

But in the opposite direction,

Note that both the numerator and denominator in the last expression we got are raised to the same power, therefore we can write the expression using the above power law as a fraction in parentheses raised to a power:
$\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}}=\big(\frac{36}{9}\big)^{\frac{1}{2}}$

We emphasize that we could do this only because both the numerator and denominator of the fraction were raised to the same power,

Let's summarize our solution steps so far we got that:

$\frac{\sqrt{36}}{\sqrt{9}}=\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}} =\big(\frac{36}{9}\big)^{\frac{1}{2}}$

Now let's calculate (by reducing the fraction) the expression inside the parentheses:

$\big(\frac{36}{9}\big)^{\frac{1}{2}} =4^\frac{1}{2}$

and we'll return to the root form using the definition of root as a power mentioned above, but in the opposite direction:

$$$$$a^{\frac{1}{n}}=\sqrt[n]{a}$

Let's apply this definition to the expression we got:

$4^\frac{1}{2}=\sqrt[2]{4}\ =\sqrt{4}=2$

where in the last step we calculated the numerical value of the root of 4,

Let's summarize the solution steps, we got that:

$\frac{\sqrt{36}}{\sqrt{9}}=\big(\frac{36}{9}\big)^{\frac{1}{2}} =\sqrt{4}=2$

Therefore the correct answer is answer B.

To solve the given problem of finding the square root of the fraction $\frac{1}{36}$, we will use the square root property for fractions. This property states that the square root of a fraction is the fraction of the square roots of the numerator and the denominator. Let's follow these steps:

• Step 1: Identify the fraction given, which is $\frac{1}{36}$.
• Step 2: Apply the square root property such that $\sqrt{\frac{1}{36}} = \frac{\sqrt{1}}{\sqrt{36}}$.
• Step 3: Calculate the square root of the numerator: $\sqrt{1} = 1$.
• Step 4: Calculate the square root of the denominator: $\sqrt{36} = 6$.
• Step 5: Form the fraction: $\frac{1}{6}$.

By following these steps, we have successfully simplified the expression. Therefore, the square root of $\frac{1}{36}$ is $\frac{1}{6}$, which matches choice 2 among the provided answer choices.

Thus, the correct and final answer to the problem $\sqrt{\frac{1}{36}} =$ is $\frac{1}{6}$.

Let's simplify the expression, first we'll reduce the fraction under the square root:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=$

We'll use two exponent laws:

A. Definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

B. The power law for powers applied to terms in parentheses:

$\big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}$

Let's return to the expression we received, first we'll use the law mentioned in A and convert the square root to a power:

$\sqrt{\frac{1}{2}}=\\ \big(\frac{1}{2}\big)^{\frac{1}{2}}=$

We'll continue and apply the power law mentioned in B, meaning- we'll apply the power separately to the numerator and denominator, in the next step we'll remember that raising the number 1 to any power will always give the result 1, and in the fraction's denominator we'll return to the root notation, again, using the power law mentioned in A (in the opposite direction):

$\big(\frac{1}{2}\big)^{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$ Let's summarize the simplification of the given expression:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$Therefore, the correct answer is answer D.

Let's simplify the expression. First, we'll reduce the fraction under the square root, then we'll calculate the result of the root:

$\sqrt{\frac{225}{25}}= \\ \sqrt{9}\\ \boxed{3}$Therefore, the correct answer is option B.