Examples with solutions for Square Root Quotient Property: Applying the formula

Exercise #1

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 #2

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 #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

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 #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

Exercise #6

Solve the following exercise:

497= \frac{\sqrt{49}}{7}=

Video Solution

Step-by-Step Solution

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 49 \sqrt{49} . Since 49 is a perfect square, the square root of 49 is 7 7 , because 7×7=49 7 \times 7 = 49 .
Step 2: Divide the result obtained in Step 1 by 7. So we have: 497=77\frac{\sqrt{49}}{7} = \frac{7}{7}

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

Therefore, the solution to the problem is 1 1 .

Answer

1 1

Exercise #7

Solve the following exercise:

14436= \sqrt{\frac{144}{36}}=

Video Solution

Step-by-Step Solution

To solve the problem 14436 \sqrt{\frac{144}{36}} , we will proceed with the following steps:

  • Step 1: Simplify the fraction 14436\frac{144}{36}. This equals to 4 because 14436=144÷3636÷36=41 \frac{144}{36} = \frac{144 \div 36}{36 \div 36} = \frac{4}{1}.
  • Step 2: Find the square root of the simplified fraction. Since 14436\frac{144}{36} simplifies to 4, we find 4\sqrt{4}.
  • Step 3: Calculate 4\sqrt{4}, which equals 2.

Therefore, the solution to the problem is 2 2 .

Answer

2 2

Exercise #8

Solve the following exercise:

644= \sqrt{\frac{64}{4}}=

Video Solution

Step-by-Step Solution

To solve the problem of finding 644 \sqrt{\frac{64}{4}} , we will proceed as follows:

  • Step 1: Simplify the fraction 644 \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 644 \frac{64}{4} . When we divide 64 by 4, we obtain 16.

So, 644=16 \frac{64}{4} = 16 .

Step 2: Calculate the square root.

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

Therefore, the solution to the problem 644 \sqrt{\frac{64}{4}} is 4.

Answer

4

Exercise #9

Solve the following exercise:

1444= \frac{\sqrt{144}}{\sqrt{4}}=

Video Solution

Step-by-Step Solution

We are tasked with solving the expression 1444 \frac{\sqrt{144}}{\sqrt{4}} . To proceed, we will use the square root quotient property.

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

1444=1444 \frac{\sqrt{144}}{\sqrt{4}} = \sqrt{\frac{144}{4}}

Next, simplify the fraction inside the square root:

1444=36 \frac{144}{4} = 36

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

36=6 \sqrt{36} = 6

Thus, the value of 1444 \frac{\sqrt{144}}{\sqrt{4}} is 6 6 .

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

Answer

6 6

Exercise #10

Solve the following exercise:

102= \frac{\sqrt{10}}{\sqrt{2}}=

Video Solution

Step-by-Step Solution

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 102=102\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{\frac{10}{2}}.
Step 2: Simplify the fraction inside the square root: 102=5\frac{10}{2} = 5.
Step 3: Therefore, 102=5\sqrt{\frac{10}{2}} = \sqrt{5}.

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

Answer

5 \sqrt{5}

Exercise #11

Solve the following exercise:

6416= \frac{\sqrt{64}}{\sqrt{16}}=

Video Solution

Step-by-Step Solution

To solve the expression 6416\frac{\sqrt{64}}{\sqrt{16}}, we will use the square root quotient property, which states:

  • ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}, assuming b0b \neq 0.

Applying this property, we have:

6416=6416\frac{\sqrt{64}}{\sqrt{16}} = \sqrt{\frac{64}{16}}.

Next, we calculate the division within the square root:

6416=4\frac{64}{16} = 4.

Therefore, we now find the square root of 4:

4=2\sqrt{4} = 2.

Hence, the result of the original expression 6416\frac{\sqrt{64}}{\sqrt{16}} is 2\mathbf{2}.

Answer

2

Exercise #12

Solve the following exercise:

644= \sqrt{\frac{64}{4}}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Simplify the fraction 644\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 644\frac{64}{4}. The division yields 1616, so we have 16\sqrt{16}.

Step 2: Using the Square Root Quotient Property, 644=644\sqrt{\frac{64}{4}} = \frac{\sqrt{64}}{\sqrt{4}}.

Step 3: Calculate the square roots: 64=8\sqrt{64} = 8 and 4=2\sqrt{4} = 2, so 82=4\frac{8}{2} = 4.

Thus, the solution to the problem is 644=4\sqrt{\frac{64}{4}} = 4.

Therefore, the correct answer is 44, which corresponds to choice 3 in the given options.

Answer

4

Exercise #13

Complete the following exercise:

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

Video Solution

Step-by-Step Solution

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 ab=ab\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 936\frac{9}{36}.
We notice that both 9 and 36 have a common factor of 9. So, we simplify:

936=14\frac{9}{36} = \frac{1}{4}

Step 2: Apply the square root quotient property:

936=14\sqrt{\frac{9}{36}} = \sqrt{\frac{1}{4}}

Now address the square root of the simplified fraction:

14=14\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}

Step 3: Calculate the square roots:

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

Step 4: Simplify the fraction:

12\frac{1}{2}

Therefore, the solution to the problem is 12\frac{1}{2}.

Answer

12 \frac{1}{2}

Exercise #14

Complete the following exercise:

10025= \sqrt{\frac{100}{25}}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Simplify the given fraction 10025\frac{100}{25}.
  • Step 2: Use the formula ab=ab\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 10025\frac{100}{25} simplifies to 44 because 100÷25=4100 \div 25 = 4.

Step 2 involves applying the square root:

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

In Step 3, calculate the square root:

4=2\sqrt{4} = 2.

Therefore, the solution to the problem is 22.

Answer

2

Exercise #15

Complete the following exercise:

819= \sqrt{\frac{81}{9}}=

Video Solution

Step-by-Step Solution

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 819\frac{81}{9}. This simplifies to 99, as dividing 8181 by 99 gives us 99.

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

Step 3: Calculate the square root. 9=3\sqrt{9} = 3, since 3×3=93 \times 3 = 9.

Therefore, the solution to the problem is 3 3 .

Answer

3

Exercise #16

Solve the following exercise:

1004= \sqrt{\frac{100}{4}}=

Video Solution

Step-by-Step Solution

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

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

Let's apply this property to the expression 1004 \sqrt{\frac{100}{4}} :

  • Step 1: Calculate 100\sqrt{100}. The square root of 100 is 10, because 10×10=10010 \times 10 = 100.
  • Step 2: Calculate 4\sqrt{4}. The square root of 4 is 2, because 2×2=42 \times 2 = 4.
  • Step 3: Divide the results from Step 1 and Step 2, using the formula:
    1004=102=5 \frac{\sqrt{100}}{\sqrt{4}} = \frac{10}{2} = 5

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

Answer

5

Exercise #17

Complete the following exercise:

19649= \sqrt{\frac{196}{49}}=

Video Solution

Step-by-Step Solution

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 19649=19649 \sqrt{\frac{196}{49}} = \frac{\sqrt{196}}{\sqrt{49}} .
Step 2: Calculate the individual square roots: 196=14 \sqrt{196} = 14 and 49=7 \sqrt{49} = 7 .
Step 3: Simplify the expression to 147=2 \frac{14}{7} = 2 .

Therefore, the solution to the problem is 2.

Answer

2

Exercise #18

Complete the following exercise:

1964= \sqrt{\frac{196}{4}}=

Video Solution

Step-by-Step Solution

To solve the problem 1964 \sqrt{\frac{196}{4}} , we can apply the following steps:

  • Step 1: Simplify the fraction 1964\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 1964\frac{196}{4} can be simplified by dividing 196 by 4.

1964=49 \frac{196}{4} = 49

Step 2: Take the square root of 49.

49=7\sqrt{49} = 7 because 7×7=497 \times 7 = 49.

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

Answer

7

Exercise #19

Complete the following exercise:

12111= \frac{\sqrt{121}}{11}=

Video Solution

Step-by-Step Solution

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:

121=11 \sqrt{121} = 11

Next, divide this result by 11:

1111=1 \frac{11}{11} = 1

Thus, the value of 12111 \frac{\sqrt{121}}{11} is 1 1 .

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

Answer

1