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.

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.

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 $\frac{1}{6}$.

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.

Express the definition of root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:

$n=2$

Meaning:

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

Let's return to the problem and convert the numerator of the fraction by using the root definition that we mentioned above :

$\frac{\sqrt{x^4}}{x}=\frac{(x^4)^{\frac{1}{2}}}{x}$

Let's recall the power law for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Apply this law to the numerator of the fraction in the expression that we obtained in the last step:

$\frac{(x^4)^{\frac{1}{2}}}{x}=\frac{x^{4\cdot\frac{1}{2}}}{x}=\frac{x^\frac{4}{2}}{x}$

In the first step we applied the above power law and in the second step we performed the multiplication in the power exponent of the numerator term,

Continue to simplify the expression that we obtained. Begin by reducing the fraction with the power exponent in the numerator term and then proceed to apply the power law for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Simplify the fraction in the now complete expression:

$\frac{x^\frac{4}{2}}{x}=\frac{x^2}{x}=x^{2-1}=x$

Let's summarize the various steps of the solution that we obtained: As shown below

$$$\frac{\sqrt{x^4}}{x}=\frac{(x^4)^{\frac{1}{2}}}{x}=\frac{x^2}{x}=x$

Therefore the correct answer is answer A.

Express the following root as a power:

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

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown

$n=2$

Meaning:

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

Let's return to the problem and use the root definition that we mentioned above to convert the root in the fraction's numerator:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}$

Remember the two following laws of exponents:

a. The law of exponents for a power applied to a product inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these laws to the fraction's numerator in the expression that we obtained in the last step:

$\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}\cdot(x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}$

$$In the first stage we applied the above-mentioned law of exponents noted in a' and then proceeded to applythe power to both factors of the product (in parentheses) in the fraction's numerator. We we careful to use parentheses given that one of the factors in the parentheses is already raised to a power.

In the second stage we applied the second law of exponents mentioned in b' to the second factor in the product,

Let's simplify the expression that we obtained:

$\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}=\frac{\sqrt{49}x^{\frac{2}{2}}}{x}=\frac{7x^1}{x}$

In the first stage we converted the fraction's power back to a root, for the first factor in the product, using the definition of root as a power mentioned at the beginning of the solution ( in the opposite direction)

Additionally- we calculated the product in the exponent of the second factor in the product in the fraction's numerator in the expression that we obtained. We then we simplified the resulting fraction.

Finish the calculation and proceed to simplify the resulting fraction:

$\frac{7x^1}{x}=\frac{7\not{x}}{\not{x}}=7$

Let's summarize the various steps of the solution that we obtained thus far, as shown below:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x} =7$

Therefore the correct answer is answer c.

Express the definition of root as a power:

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

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

$n=2$

meaning:

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

Proceed to convert the expression using the root definition we mentioned above:

$\frac{\sqrt{25x^2}}{\sqrt{x^2}}=\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}}$

Now let's recall two laws of exponents:

a. The law of exponents for a power applied to a product inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these laws to the numerator and denominator of the fraction in the expression that we obtained in the last step:

$\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}}=\frac{25^{\frac{1}{2}}\cdot(x^2)^{\frac{1}{2}}}{(x^2)^{\frac{1}{2}}}=\frac{25^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x^{2\cdot\frac{1}{2}}}$

In the first stage we applied the above law of exponents mentioned in a' and then proceeded to apply the power to both factors of the product inside of the parentheses in the fraction's numerator.

We carried this out carefully by using parentheses given that one of the factors in the parentheses is already raised to a power. In the second stage we applied the second law of exponents mentioned in b' to the second factor in the product in the fraction's numerator and similarly to the factor in the fraction's denominator,

Let's simplify all of the expressions that we obtained:

$\frac{25^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x^{2\cdot\frac{1}{2}}}=\frac{\sqrt{25}x^{\frac{2}{2}}}{x^{\frac{2}{2}}}=\frac{5x^1}{x^1}$

In the first stage we converted the fraction's power back to a root. For the first factor in the product, this was done using the definition of root as a power mentioned at the beginning of the solution (in the opposite direction)

We then proceeded to calculate the numerical value of the root.

Additionally - we calculated the product of the power of the second factor in the product in the fraction's numerator in the expression that we obtained. Similarly we carried this out for the factor in the fraction's denominator. We then simplified the resulting fraction for that factor.

Let's complete the calculation and simplify the resulting fraction:

$\frac{5x^1}{x^1} =\frac{5\not{x}}{\not{x}}=5$

Let's summarize the steps of the solution thus far, as seen below:

$\frac{\sqrt{25x^2}}{\sqrt{x^2}}=\frac{(25x^2)^{\frac{1}{2}}}{(x^{2)^{\frac{1}{2}}}} =\frac{5x^1}{x^1} =5$

Therefore the correct answer is answer c.

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

• Step 1: Calculate the square root of 49.
• Step 2: Divide the result by 7.

Now, let's work through each step:
Step 1: Calculate the square root of 49. We have $\sqrt{49}$. Since 49 is a perfect square, the square root of 49 is $7$, because $7 \times 7 = 49$.
Step 2: Divide the result obtained in Step 1 by 7. So we have: $\frac{\sqrt{49}}{7} = \frac{7}{7}$

The result of this division is $1$, because dividing any number by itself (except zero) yields 1.

Therefore, the solution to the problem is $1$.

To solve the problem $\sqrt{\frac{144}{36}}$, we will proceed with the following steps:

• Step 1: Simplify the fraction $\frac{144}{36}$. This equals to 4 because $\frac{144}{36} = \frac{144 \div 36}{36 \div 36} = \frac{4}{1}$.
• Step 2: Find the square root of the simplified fraction. Since $\frac{144}{36}$ simplifies to 4, we find $\sqrt{4}$.
• Step 3: Calculate $\sqrt{4}$, which equals 2.

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

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.

We are tasked with solving the expression $\frac{\sqrt{144}}{\sqrt{4}}$. To proceed, we will use the square root quotient property.

According to the square root quotient property, $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. Applying this to the given expression, we have:

Next, simplify the fraction inside the square root:

Now, we need to find the square root of 36:

Thus, the value of $\frac{\sqrt{144}}{\sqrt{4}}$ is $6$.

Therefore, the solution to the problem is $\boxed{6}$.

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

• Step 1: Apply the square root quotient property.
• Step 2: Simplify the fraction under the square root.
• Step 3: Evaluate the square root if possible.

Now, let's work through each step:
Step 1: The square root quotient property tells us that $\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}}$.
Step 2: Simplify the fraction inside the square root: $\frac{10}{2} = 5$.
Step 3: Therefore, $\sqrt{\frac{10}{2}} = \sqrt{5}$.

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

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 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.