Square Root Quotient Property: Number of terms

To solve the given exercise, let's simplify each square root expression separately:

• Step 1: Simplify $\sqrt{\frac{2}{4}}$.

  The fraction $\frac{2}{4}$ simplifies to $\frac{1}{2}$. Thus, $\sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}}$.

• Step 2: Simplify $\sqrt{\frac{8}{16}}$.

  The fraction $\frac{8}{16}$ simplifies to $\frac{1}{2}$. Thus, $\sqrt{\frac{8}{16}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}}$.

• Step 3: Multiply the results from Step 1 and Step 2.

  $\sqrt{\frac{2}{4}} \cdot \sqrt{\frac{8}{16}} = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}} = \frac{1}{2}$.

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

To solve this problem, we'll simplify the given expression step by step:

First, let's simplify the numerator:

$\sqrt{10} \cdot \sqrt{5} \cdot \sqrt{2} = \sqrt{10 \cdot 5 \cdot 2} = \sqrt{100}$.

Simplifying further, $\sqrt{100} = 10$.

Next, simplify the denominator:

$\sqrt{5} \cdot \sqrt{5} \cdot \sqrt{4} = \sqrt{5 \cdot 5 \cdot 4} = \sqrt{100}$.

And $\sqrt{100} = 10$.

Now, divide the simplified numerator by the simplified denominator:

$\frac{10}{10} = 1$.

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

To solve the expression $\sqrt{\frac{2}{4}} \cdot \sqrt{6}$, we will break it down and simplify step by step.

Step 1: Simplify the square root of the fraction.
$\sqrt{\frac{2}{4}}$ can be rewritten using the square root of a quotient property:
$\sqrt{\frac{2}{4}} = \frac{\sqrt{2}}{\sqrt{4}}$.

Step 2: Simplify $\sqrt{4}$.
Since $\sqrt{4} = 2$, the expression becomes:
$\frac{\sqrt{2}}{2}$.

Step 3: Multiply by $\sqrt{6}$.
Now multiply $\frac{\sqrt{2}}{2}$ by $\sqrt{6}$:
$\frac{\sqrt{2}}{2} \cdot \sqrt{6} = \frac{\sqrt{2 \cdot 6}}{2}$.

Step 4: Simplify the square root.
The multiplication inside the square root becomes $\sqrt{12}$, so:
$\frac{\sqrt{12}}{2}$.

Step 5: Simplify $\sqrt{12}$.
Since $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$,
this results in $\frac{2\sqrt{3}}{2} = \sqrt{3}$.

Therefore, the solution to the problem is $\sqrt{3}$.

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

• Simplify the numerator by multiplying the square roots together.
• Simplify the denominator by recognizing that $\sqrt{2} \times \sqrt{2}$ equals 2.
• Apply the quotient rule for square roots to simplify the expression.

Now, let's work through each step:
Step 1: The numerator is $\sqrt{12} \cdot \sqrt{4} \cdot \sqrt{3}$. Using the product property, combine them into one square root:
$\sqrt{12 \times 4 \times 3} = \sqrt{144}$.

Step 2: The denominator is $\sqrt{2} \cdot \sqrt{2} = \sqrt{4} = 2$.

Step 3: Now apply the quotient rule:
$\frac{\sqrt{144}}{2} = \sqrt{\frac{144}{4}} = \sqrt{36}$.

The result of $\sqrt{36}$ is 6.

Therefore, the solution to the problem is $6$.

To solve the problem, we'll simplify the expression $\frac{\sqrt{81}\cdot\sqrt{4}}{\sqrt{9}\cdot\sqrt{9}}$ using square root properties and arithmetic operations.

Step 1: Simplify each square root individually:

• $\sqrt{81} = 9$, $\sqrt{4} = 2$.
• Both $\sqrt{9}$ terms are equal to 3.

Thus, the expression becomes $\frac{9 \cdot 2}{3 \cdot 3}$.

Step 2: Perform multiplication of numbers:

• Numerator: $9 \times 2 = 18$.
• Denominator: $3 \times 3 = 9$.

The expression is now $\frac{18}{9}$.

Step 3: Simplify the fraction:

• $\frac{18}{9} = 2$, after dividing both the numerator and the denominator by the GCD, which is 9.

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

To solve this problem, let's simplify each term in the expression step-by-step:

• Simplify the first term $\frac{\sqrt{36}}{\sqrt{2}\cdot\sqrt{3}}$:

  • $\sqrt{36} = 6$, as 36 is a perfect square.
  • Apply the property of square roots: $\sqrt{2} \cdot \sqrt{3} = \sqrt{6}$.
  • Rewrite the expression: $\frac{6}{\sqrt{6}} = \frac{6}{\sqrt{6}}$.
  • Using the square root quotient property: $\frac{6}{\sqrt{6}} = \sqrt{\frac{6^2}{6}} = \sqrt{6}$.

• Simplify the second term $\frac{\sqrt{25}}{5}$:

  • $\sqrt{25} = 5$, as 25 is a perfect square.
  • The expression becomes $\frac{5}{5} = 1$.

• Combine the simplified terms: $\sqrt{6} + 1$

Therefore, the solution to the problem is $\sqrt{6} + 1$.

To solve this problem, we'll simplify the expression step-by-step:

Step 1: Simplify each root expression:
- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$
- $\sqrt{16} = 4$
- $\sqrt{64} = 8$
- $\sqrt{2} = \sqrt{2}$
- $\sqrt{4} = 2$

Step 2: Substitute back into the original expression:
$\frac{\sqrt{8}}{2 \cdot \sqrt{16}} = \frac{2\sqrt{2}}{2 \cdot 4} = \frac{\sqrt{2}}{4}$

$\frac{\sqrt{64}}{\sqrt{2} \cdot \sqrt{4}} = \frac{8}{\sqrt{2} \cdot 2} = \frac{8}{2\sqrt{2}} = \frac{4}{\sqrt{2}}$

Step 3: Multiply the simplified fractions:
$\frac{\sqrt{2}}{4} \cdot \frac{4}{\sqrt{2}} = \frac{\sqrt{2} \cdot 4}{4 \cdot \sqrt{2}} = \frac{4\sqrt{2}}{4\sqrt{2}} = 1$

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

To solve the expression $\frac{\sqrt{2}\cdot\sqrt{8}}{\sqrt{64}}+\frac{\sqrt{4}\cdot\sqrt{4}}{\sqrt{4}\cdot\sqrt{16}}$, let's simplify each term step-by-step:

First, consider the term $\frac{\sqrt{2}\cdot\sqrt{8}}{\sqrt{64}}$:

• Simplify $\sqrt{2} \cdot \sqrt{8}$ using the product property: $\sqrt{2 \cdot 8} = \sqrt{16}$.
• We know that $\sqrt{16} = 4$.
• $\sqrt{64} = 8$.
• Thus, $\frac{\sqrt{16}}{\sqrt{64}}$ becomes $\frac{4}{8} = \frac{1}{2}$.

Next, consider the term $\frac{\sqrt{4}\cdot\sqrt{4}}{\sqrt{4}\cdot\sqrt{16}}$:

• Simplify $\sqrt{4} \cdot \sqrt{4}$ using the product property: $\sqrt{4 \cdot 4} = \sqrt{16}$.
• We know that $\sqrt{16} = 4$.
• Simplify the denominator $\sqrt{4} \cdot \sqrt{16}$ using the product property: $\sqrt{4 \cdot 16} = \sqrt{64}$, which is $8$.
• Thus, $\frac{\sqrt{16}}{\sqrt{64}}$ becomes $\frac{4}{8} = \frac{1}{2}$.

Finally, add the simplified terms together:

$\frac{1}{2} + \frac{1}{2} = 1$.

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

To solve this problem, we'll proceed with the following steps:

• Step 1: Simplify $\sqrt{\frac{49}{4}}$.
• Step 2: Simplify $\sqrt{\frac{9}{36}}$.
• Step 3: Add the results obtained from Step 1 and Step 2.

Step 1: We use the square root of a quotient property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. For $\sqrt{\frac{49}{4}}$:

$\sqrt{\frac{49}{4}} = \frac{\sqrt{49}}{\sqrt{4}} = \frac{7}{2}$

Step 2: Similarly, apply the same property to $\sqrt{\frac{9}{36}}$:

$\sqrt{\frac{9}{36}} = \frac{\sqrt{9}}{\sqrt{36}} = \frac{3}{6} = \frac{1}{2}$

Step 3: Add the two results obtained:

$\frac{7}{2} + \frac{1}{2} = \frac{7 + 1}{2} = \frac{8}{2} = 4$

Therefore, the solution to the problem is $4$.

To solve this problem, let's follow these detailed steps:

The given expression is:
$\frac{\sqrt{2}\cdot\sqrt{4}}{\sqrt{144}+\sqrt{16}}$

Step 1: Simplify the numerator.

In the numerator, we have $\sqrt{2} \cdot \sqrt{4}$. Using the property of square roots, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, we can write:

• $\sqrt{2} \cdot \sqrt{4} = \sqrt{2 \cdot 4} = \sqrt{8}$

We can simplify $\sqrt{8}$ further:

• $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

Step 2: Simplify the denominator.

In the denominator, we have $\sqrt{144} + \sqrt{16}$. Let's compute each square root:

• $\sqrt{144} = 12$ because $12^2 = 144$
• $\sqrt{16} = 4$ because $4^2 = 16$

Thus, the denominator becomes:

• $12 + 4 = 16$

Step 3: Form the fraction and simplify it.

Replacing the simplified numerator and denominator, the expression becomes:

• $\frac{2\sqrt{2}}{16}$

Simplifying the fraction, divide both terms in the fraction by 2:

• $\frac{2\sqrt{2}}{16} = \frac{\sqrt{2}}{8}$

Therefore, the solution to the problem is:

$\frac{\sqrt{2}}{8}$

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

• Identify and combine the square roots into a single square root.
• Simplify the expression logically inside the square root.
• Resolve the final numerical value.

Now, let's work through each step:

Step 1: Convert the expression:

$\sqrt{\frac{3}{2}} \cdot \sqrt{3} \cdot \sqrt{\frac{9}{2}} = \sqrt{\frac{3}{2} \cdot 3 \cdot \frac{9}{2}}$

This results in a single square root.

Step 2: Simplify inside the square root:

$\frac{3}{2} \cdot 3 \cdot \frac{9}{2} = \frac{3 \cdot 3 \cdot 9}{2 \cdot 2} = \frac{81}{4}$

Step 3: Calculate the square root:

• The square root of $\frac{81}{4}$ can be found by separately taking square roots of the numerator and the denominator:
• $\sqrt{\frac{81}{4}} = \frac{\sqrt{81}}{\sqrt{4}} = \frac{9}{2}$

Thus, combining all parts logically, we resolve the expression:

$\frac{9}{2} = 4\frac{1}{2}$

Therefore, the solution to the problem is $\boxed{4\frac{1}{2}}$.

To solve this problem, we'll eliminate fractions under the square roots by simplifying directly:

• Step 1: Simplify each fraction inside the square roots:
  $\frac{4}{2} = 2$ and $\frac{6}{3} = 2$.
• Step 2: Apply the square root to each simplified fraction:
  $\sqrt{\frac{4}{2}} = \sqrt{2}$ and $\sqrt{\frac{6}{3}} = \sqrt{2}$.
• Step 3: Add the results of the square roots:
  $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$.

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