Square Root Quotient Property - Examples, Exercises and 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.

To solve the problem, we'll take the following steps:

• Compute the square root of the number 121.
• Divide the square root result by 11.

Let's go through the calculations:

First, compute the square root of 121:

$\sqrt{121} = 11$

Next, divide this result by 11:

$\frac{11}{11} = 1$

Thus, the value of $\frac{\sqrt{121}}{11}$ is $1$.

Therefore, the correct answer is choice 1, which corresponds to 1.

To solve the problem $\sqrt{\frac{196}{4}}$, we can apply the following steps:

• Step 1: Simplify the fraction $\frac{196}{4}$.
• Step 2: Compute the square root of the result from Step 1.

Let's go through these steps in detail:

Step 1: Simplify the fraction.

The fraction $\frac{196}{4}$ can be simplified by dividing 196 by 4.

$\frac{196}{4} = 49$

Step 2: Take the square root of 49.

$\sqrt{49} = 7$ because $7 \times 7 = 49$.

Therefore, the solution to the problem is $\boxed{7}$.

To solve this problem, we'll follow these steps:

• Step 1: Apply the square root quotient property.
• Step 2: Calculate the square roots of the numerator and the denominator.
• Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: Apply the square root quotient property $\sqrt{\frac{196}{49}} = \frac{\sqrt{196}}{\sqrt{49}}$.
Step 2: Calculate the individual square roots: $\sqrt{196} = 14$ and $\sqrt{49} = 7$.
Step 3: Simplify the expression to $\frac{14}{7} = 2$.

Therefore, the solution to the problem is 2.

To solve this problem, we'll apply the Square Root Quotient Property to the given expression. The property states that:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Let's apply this property to the expression $\sqrt{\frac{100}{4}}$:

• Step 1: Calculate $\sqrt{100}$. The square root of 100 is 10, because $10 \times 10 = 100$.
• Step 2: Calculate $\sqrt{4}$. The square root of 4 is 2, because $2 \times 2 = 4$.
• Step 3: Divide the results from Step 1 and Step 2, using the formula:
  $\frac{\sqrt{100}}{\sqrt{4}} = \frac{10}{2} = 5$

Therefore, the solution to the problem is $\boxed{5}$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the fraction inside the square root.
• Step 2: Apply the square root quotient property.
• Step 3: Perform the necessary calculations and simplify if needed.

Let's work through each step:

Step 1: Simplify the fraction $\frac{81}{9}$. This simplifies to $9$, as dividing $81$ by $9$ gives us $9$.

Step 2: Apply the square root property. We need to calculate $\sqrt{9}$.

Step 3: Calculate the square root. $\sqrt{9} = 3$, since $3 \times 3 = 9$.

Therefore, the solution to the problem is $3$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the fraction within the square root, if possible.
• Step 2: Apply the square root quotient property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
• Step 3: Calculate the square roots of the resulting numbers.
• Step 4: Simplify the fraction, if necessary.

Now, let's work through each step:

Step 1: Simplify the fraction $\frac{9}{36}$.
We notice that both 9 and 36 have a common factor of 9. So, we simplify:

$\frac{9}{36} = \frac{1}{4}$

Step 2: Apply the square root quotient property:

$\sqrt{\frac{9}{36}} = \sqrt{\frac{1}{4}}$

Now address the square root of the simplified fraction:

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

Step 3: Calculate the square roots:

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

Step 4: Simplify the fraction:

$\frac{1}{2}$

Therefore, the solution to the problem is $\frac{1}{2}$.

To solve this problem, let's apply the following approach:

• Step 1: Simplify the given fraction $\frac{100}{25}$.
• Step 2: Use the formula $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to find the square root of the simplified fraction.
• Step 3: Calculate the square root to arrive at the final answer.

Let's start with Step 1:

The fraction $\frac{100}{25}$ simplifies to $4$ because $100 \div 25 = 4$.

Step 2 involves applying the square root:

We can write this as $\sqrt{4}$.

In Step 3, calculate the square root:

$\sqrt{4} = 2$.

Therefore, the solution to the problem is $2$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the fraction $\frac{64}{4}$.
• Step 2: Apply the Square Root Quotient Property.
• Step 3: Calculate the square roots of the numerator and the denominator.

Now, let's work through each step:

Step 1: Simplify the fraction $\frac{64}{4}$. The division yields $16$, so we have $\sqrt{16}$.

Step 2: Using the Square Root Quotient Property, $\sqrt{\frac{64}{4}} = \frac{\sqrt{64}}{\sqrt{4}}$.

Step 3: Calculate the square roots: $\sqrt{64} = 8$ and $\sqrt{4} = 2$, so $\frac{8}{2} = 4$.

Thus, the solution to the problem is $\sqrt{\frac{64}{4}} = 4$.

Therefore, the correct answer is $4$, which corresponds to choice 3 in the given options.

To solve the expression $\frac{\sqrt{64}}{\sqrt{16}}$, we will use the square root quotient property, which states:

• $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, assuming $b \neq 0$.

Applying this property, we have:

$\frac{\sqrt{64}}{\sqrt{16}} = \sqrt{\frac{64}{16}}$.

Next, we calculate the division within the square root:

$\frac{64}{16} = 4$.

Therefore, we now find the square root of 4:

$\sqrt{4} = 2$.

Hence, the result of the original expression $\frac{\sqrt{64}}{\sqrt{16}}$ is $\mathbf{2}$.

To solve the problem of finding $\sqrt{\frac{64}{4}}$, we will proceed as follows:

• Step 1: Simplify the fraction $\frac{64}{4}$.
• Step 2: Calculate the square root of the simplified result.

Let's work through these steps:

Step 1: Simplify the fraction.

The fraction given is $\frac{64}{4}$. When we divide 64 by 4, we obtain 16.

So, $\frac{64}{4} = 16$.

Step 2: Calculate the square root.

Now, we need to find $\sqrt{16}$. We know that the square root of 16 is 4 because $4 \times 4 = 16$.

Therefore, the solution to the problem $\sqrt{\frac{64}{4}}$ is 4.