Square Roots: Solving the problem

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

• Step 1: Recognize what finding a square root means
• Step 2: List known perfect squares to identify which one results in 64
• Step 3: Verify the square root by calculation

Now, let's work through each step:
Step 1: To find the square root of 64, we seek a number that, when multiplied by itself, equals 64.
Step 2: Consider the sequence of perfect squares: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, $6^2 = 36$, $7^2 = 49$, $8^2 = 64$.
Step 3: We see that $8^2 = 64$. Therefore, the square root of 64 is 8.

Therefore, the solution to this problem is $8$.

Let's solve the problem step by step:

• Step 1: Understand what the square root means.

The square root of a number $n$ is a value that, when multiplied by itself, equals $n$. This is written as $x = \sqrt{n}$.

• Step 2: Apply this definition to the number $36$.

We are looking for a number $x$ such that $x^2 = 36$. This translates to finding $x = \sqrt{36}$.

• Step 3: Determine the correct number.

We know that $6 \times 6 = 36$. Therefore, the principal square root of $36$ is $6$.

Thus, the solution to the problem is $\sqrt{36} = 6$.

Among the given choices, the correct one is: Choice 1: $6$.

The root of 441 is 21.

$21\times21=$

$21\times20+21=$

$420+21=441$

To solve the expression $5+\sqrt{36}-1=$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Here are the steps:

First, calculate the square root:

$\sqrt{36} = 6$

Substitute the square root back into the expression:

$5 + 6 - 1$

Next, perform the addition and subtraction from left to right:

Add 5 and 6:

$5 + 6 = 11$

Then subtract 1:

$11 - 1 = 10$

Finally, you obtain the solution:

$10$

To solve this problem, we need to find the square root of $0.25$. We will follow these steps:

• Step 1: Convert $0.25$ to a more manageable form: $0.25 = \frac{25}{100} = \frac{1}{4}$.
• Step 2: Use the property of square roots: If $\sqrt{\frac{1}{4}} = y$, then $y^2 = \frac{1}{4}$.

Now, let's calculate the square root:
Since $\frac{1}{4}$ is a fraction, taking the square root of a fraction involves taking the square root of the numerator and the square root of the denominator separately:
$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.

Verify by squaring the result: $(\frac{1}{2})^2 = \frac{1}{4}$. This matches the original number, confirming our result is correct.

Therefore, the square root of $0.25$ is $0.5$.

To solve the expression $81+\sqrt{81}+10$, we need to follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Here, the expression contains a square root, which is a type of exponent operation. Therefore, we will handle the square root first:

• Find the square root of 81, which is calculated as follows: $\sqrt{81} = 9$.

Now substitute the result back into the original expression:

$81 + 9 + 10$

Next, perform the addition operations from left to right:

• First, add 81 and 9: $81 + 9 = 90$.
• Then, add the result to 10: $90 + 10 = 100$.

Therefore, the final result of the expression $81+\sqrt{81}+10$ is:

$100$

To solve the expression $143-\sqrt{121}+18$, we need to follow the order of operations, which dictate that we should simplify any expressions under a square root first, followed by subtraction and addition.

Step 1: Simplify the square root:

• Calculate the square root: $\sqrt{121}$.
• The square root of 121 is 11, because $11 \times 11 = 121$.

Now, substitute back into the expression:

• The expression becomes: $143 - 11 + 18$.

Step 2: Perform the subtraction:

• Calculate $143 - 11$.
• This equals 132, because subtracting 11 from 143 yields 132.

Step 3: Perform the addition:

• Now add 18 to the result of the subtraction: $132 + 18$.
• The result is 150, because adding 18 to 132 equals 150.

Therefore, the final answer is $150$.

To solve for the square root of $272\frac{1}{4}$, we can follow these steps:

• Step 1: Convert the mixed number to an improper fraction. $272\frac{1}{4}$ becomes $\frac{1089}{4}$.
• Step 2: Find the square root of $\frac{1089}{4}$.

Step 1: Convert the mixed number to an improper fraction:
To convert $272\frac{1}{4}$ to an improper fraction:

• Multiply the whole number 272 by 4: $272 \times 4 = 1088$
• Add the numerator of the fraction: $1088 + 1 = 1089$
• Thus, $272\frac{1}{4}$ is $\frac{1089}{4}$.

Step 2: Calculate the square root of $\frac{1089}{4}$:
The square root of a fraction $\frac{a}{b}$ is $\frac{\sqrt{a}}{\sqrt{b}}$.

• $\sqrt{\frac{1089}{4}} = \frac{\sqrt{1089}}{\sqrt{4}}$
• The square root of 1089 is 33, since $33 \times 33 = 1089$.
• The square root of 4 is 2, since $2 \times 2 = 4$.

Therefore, $\sqrt{\frac{1089}{4}} = \frac{33}{2} = 16.5 = 16\frac{1}{2}$.

Thus, the square root of $272\frac{1}{4}$ is $16\frac{1}{2}$.

According to the order of operations, we should first solve the expression inside of the parentheses:

$(\sqrt{380.25}-\frac{1}{2})=(19.5-\frac{1}{2})=(19)$

In the next step, we will proceed to solve the exponentiation, and finally the subtraction:

$(19)^2-11=(19\times19)-11=361-11=350$

To solve the expression $4\times\sqrt{0.49}+4^2 =$, we will follow the order of operations, which prioritizes operations inside parentheses, exponents and roots, followed by multiplication and division, and finally addition and subtraction.

1. Calculate the Square Root:
The first step is to solve the square root part of the expression. $\sqrt{0.49}$.
$0.49$ is a simple decimal number whose square root is $0.7$, because $0.7 \times 0.7 = 0.49$.
So,$\sqrt{0.49} = 0.7$.

2. Multiply:
Next, we multiply the result of the square root by 4:
$4 \times 0.7 = 2.8$.

3. Calculate the Power:
Evaluate $4^2$.
$4^2 = 16$, because $4 \times 4 = 16$.

4. Addition:
Now, add the results from the previous steps:
$2.8 + 16 = 18.8$.

The final result of the expression $4\times\sqrt{0.49}+4^2$ is $\boxed{18.8}$.

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

• Step 1: Calculate $\sqrt{961}$
• Step 2: Calculate $\sqrt{1}$
• Step 3: Subtract the square roots obtained in Steps 1 and 2

Now, let's work through each step:
Step 1: We need to find $\sqrt{961}$. Recognize that 961 is a perfect square and $\sqrt{961} = 31$ because $31^2 = 961$.
Step 2: Next, calculate $\sqrt{1}$. Since 1 is also a perfect square, $\sqrt{1} = 1$.
Step 3: Subtract these values: $31 - 1 = 30$.

Therefore, the solution to the problem is $30$.

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

• Step 1: Find $\sqrt{400}$.
• Step 2: Find $\sqrt{225}$.
• Step 3: Subtract $\sqrt{225}$ from $\sqrt{400}$.

Now, let's work through each step:
Step 1: Calculate $\sqrt{400}$. Since $400 = 20^2$, it follows that $\sqrt{400} = 20$.

Step 2: Calculate $\sqrt{225}$. Since $225 = 15^2$, it follows that $\sqrt{225} = 15$.

Step 3: Subtract $\sqrt{225}$ from $\sqrt{400}$:

$20 - 15 = 5$.

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

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

• Step 1: Calculate the square root of 144.
• Step 2: Add 12 to the result of the square root.

Now, let's work through each step:
Step 1: The square root of 144 is $\sqrt{144} = 12$, since 12 is the number that, when squared, gives 144.
Step 2: Add 12 to this result, which is $12 + 12 = 24$.

Therefore, the solution to the problem is $24$.

To solve this problem, we will find the square roots of the given numbers and add the results:

• First, compute $\sqrt{49}$. Since $7^2 = 49$, we have $\sqrt{49} = 7$.
• Next, compute $\sqrt{36}$. Since $6^2 = 36$, we have $\sqrt{36} = 6$.
• The final step is to sum these square roots: $7 + 6 = 13$.

Therefore, the solution to the problem is $13$.

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

• Step 1: Calculate $\sqrt{16}$.
• Step 2: Calculate $\sqrt{4}$.
• Step 3: Add the results of these operations.

Now, let's work through each step:

Step 1: Calculate the square root of 16.
Since 16 is a perfect square, $\sqrt{16} = 4$.

Step 2: Calculate the square root of 4.
Since 4 is also a perfect square, $\sqrt{4} = 2$.

Step 3: Add the results from steps 1 and 2.
Thus, $4 + 2 = 6$.

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