Square Root Quotient Property Practice Problems & Solutions

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.

Simplify the following expression:

Begin by reducing the fraction under the square root:

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

Apply 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 that we obtained. Apply the law mentioned in A and convert the square root to a power:

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

Next use the power law mentioned in B, apply the power separately to the numerator and denominator.

In the next step remember that raising the number 1 to any power will always result in 1.

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.

In order to determine the square root of the following fraction $\frac{1}{36}$, we will apply 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 given fraction, which is $\frac{1}{36}$.

• Step 2: Apply the square root property as follows $\sqrt{\frac{1}{36}} = \frac{\sqrt{1}}{\sqrt{36}}$.

• 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 $\frac{1}{6}$.

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

Express the definition of root as a power:

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

$$Remember that for a square root (also called "root to the power of 2") we don't write the root's power:

$n=2$

meaning:

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

Thus we will proceed to convert all the roots in the problem to powers:

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

Below is the power law for a fraction inside of parentheses:

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

However in the opposite direction,

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

We can only do this because both the numerator and denominator of the fraction were raised to the same power,

Let's summarize the different steps of our solution so far:

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

Proceed to calculate (by reducing the fraction) the expression inside of 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, ( however this time in the opposite direction):

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

Let's apply this definition to the expression that we obtained:

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

Once in the last step we calculate the numerical value of the root of 4,

To summarize we obtained the following calculation: :

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

Therefore the correct answer is answer B.

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.