Square root of a quotient

🏆Practice square root quotient property

Root of a quotient

When we find a root that is in the complete quotient (in the complete fraction), we can break down the factors of the quotient: the numerator and the denominator and leave the root separated for each of them. We will not forget to leave the division symbol: the dividing line between the factors we separate.

Let's put it this way:

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

Start practice

Test yourself on square root quotient property!

einstein

Solve the following exercise:

\( \sqrt{\frac{2}{4}}= \)

Practice more now

Let's look at this in the example

10064\sqrt{\frac{100}{64}}
According to the quotient root rule, we can break down the factors and leave the root of each factor separately while maintaining the multiplication operation between them:
We will break it down and obtain:
10064\frac{\sqrt{100}}{\sqrt{64}}

10×8=8010\times 8=80


If you are interested in this article, you might also be interested in the following articles:

Laws of Radicals

The Root of a Product

Radication

Combining Powers and Roots

In the blog of Tutorela you will find a variety of articles about mathematics.


Examples and exercises with solutions of the root of the quotient

Exercise #1

Solve the following exercise:

24= \sqrt{\frac{2}{4}}=

Video Solution

Step-by-Step Solution

Simplify the following expression:

Begin by reducing the fraction under the square root:

24=12= \sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=

Apply two exponent laws:

A. Definition of root as a power:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

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

(ab)n=anbn \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:

12=(12)12= \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):

(12)12=112212=12 \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:

24=12=112212=12 \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.

Answer

12 \frac{1}{\sqrt{2}}

Exercise #2

Complete the following exercise:

136= \sqrt{\frac{1}{36}}=

Video Solution

Step-by-Step Solution

In order to determine the square root of the following fraction 136\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 136\frac{1}{36}.

  • Step 2: Apply the square root property as follows 136=136\sqrt{\frac{1}{36}} = \frac{\sqrt{1}}{\sqrt{36}}.

  • Step 3: Calculate the square root of the numerator: 1=1\sqrt{1} = 1.

  • Step 4: Calculate the square root of the denominator: 36=6\sqrt{36} = 6.

  • Step 5: Form the fraction: 16\frac{1}{6}.

By following these steps, we have successfully simplified the expression. Therefore, the square root of 136\frac{1}{36} is 16\frac{1}{6}.

Thus, the correct and final answer to the problem 136= \sqrt{\frac{1}{36}} = is 16\frac{1}{6}.

Answer

16 \frac{1}{6}

Exercise #3

Solve the following exercise:

369= \frac{\sqrt{36}}{\sqrt{9}}=

Video Solution

Step-by-Step Solution

Express the definition of root as a power:

an=a1n \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 n=2

meaning:

a=a2=a12 \sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}

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

369=3612912 \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:

ancn=(ac)n \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:
3612912=(369)12 \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:

369=3612912=(369)12 \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:

(369)12=412 \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):

a1n=an a^{\frac{1}{n}}=\sqrt[n]{a}

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

412=42 =4=2 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: :

369=(369)12=4=2 \frac{\sqrt{36}}{\sqrt{9}}=\big(\frac{36}{9}\big)^{\frac{1}{2}} =\sqrt{4}=2

Therefore the correct answer is answer B.

Answer

2 2

Exercise #4

Choose the expression that is equal to the following:

a:b \sqrt{a}:\sqrt{b}

Video Solution

Step-by-Step Solution

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

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

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

Answer

a:b \sqrt{a:b}

Exercise #5

Solve the following exercise:

22525= \sqrt{\frac{225}{25}}=

Video Solution

Step-by-Step Solution

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

22525=93 \sqrt{\frac{225}{25}}= \\ \sqrt{9}\\ \boxed{3} Therefore, the correct answer is option B.

Answer

3

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge
Start practice