Square Roots - Examples, Exercises and Solutions

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$We can determine that:

5>4>3>1 Therefore, the correct answer is option A

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: $6$.

To solve this problem, we follow these steps:

• Step 1: Understand that finding the square root of a number means determining what number, when multiplied by itself, equals the original number.
• Step 2: Identify the numbers that could potentially be the square root of $49$. These are $\pm7$, but by convention, the square root function typically refers to the non-negative root.
• Step 3: Calculate $7 \times 7 = 49$. This confirms that $\sqrt{49} = 7$.
• Step 4: Verify using the problem's multiple-choice answers to ensure $7$ is among them, confirming choice number .

Therefore, the solution to the problem $\sqrt{49}$ is $7$.

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 $8$.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for power of a power:

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

Let's start with converting the square root to an exponent using the law mentioned in a':

$\sqrt{x^2}= \\ \downarrow\\ (x^2)^{\frac{1}{2}}=$We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$(x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x}$Therefore, the correct answer is answer a'.

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:

$10$

The task is to find the square root of the number 100. The square root operation seeks a number which, when squared, equals the original number. For any positive integer, if $x^2 = 100$, then $x$ should be our answer.

Step 1: Recognize that 100 is a perfect square. This means there exists an integer $x$ such that $x \times x = 100$. Generally, we recall basic squares such as:

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• and so forth, up to $10^2$

Step 2: Checking integers, we find that:

$10^2 = 10 \times 10 = 100$

Step 3: Confirm the result: Since $10 \times 10 = 100$, then $\sqrt{100} = 10$.

Step 4: Compare with answer choices. Given that one of the choices is 10, and $\sqrt{100} = 10$, choice 1 is correct.

Therefore, the square root of 100 is 10.

To solve the problem of finding the square root of 121, let's follow these steps:

• Step 1: Understand that we need to find a number $x$ such that $x^2 = 121$.
• Step 2: Recognize that 121 is a perfect square. Specifically, $11 \times 11 = 121$.
• Step 3: Therefore, the square root of 121 is clearly $11$.

Thus, the solution to the problem is $11$.

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

• Step 1: Identify that the problem asks for the square root of 144.
• Step 2: Use the concept of square roots, where $\sqrt{n} = x$ implies $x^2 = n$.
• Step 3: Check for a perfect square whose square equals 144.

Now, let's solve the problem:

Step 1: We need the square root $\sqrt{144}$.

Step 2: Recall that $x^2 = 144$. We need to find $x$.

Step 3: Recognize that 144 is a perfect square and find $x$ such that $x^2 = 144$. Through either calculation or prior knowledge, we know:
$12^2 = 144$.

Therefore, the square root of 144 is $\sqrt{144} = 12$.

Thus, the solution to the problem is $\mathbf{12}$.

To determine the square root of 16, follow these steps:

• Identify that we are looking for the square root of 16, which is a number that, when multiplied by itself, equals 16.
• Recall the basic property of perfect squares: $4 \times 4 = 16$.
• Thus, the square root of 16 is 4.

Hence, the solution to the problem is the principal square root, which is $4$.

To solve the problem, we will follow these steps:

• Step 1: Identify the perfect squares near 225.
• Step 2: Calculate to find which integer when squared equals 225.

Now, let's work through the steps:

Step 1: The perfect squares around 225 are $196 = 14^2$, $225 = 15^2$, and $256 = 16^2$.
Step 2: We calculate $15^2 = 15 \times 15 = 225$, therefore, $\sqrt{225} = 15$.

Therefore, the solution to the problem is $\sqrt{225} = 15$.

Accordingly, the correct answer choice is option 2: 15.

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

• Step 1: Understand the definition of a square root.
• Step 2: Identify which integer, when squared, gives 36.
• Step 3: Verify this integer meets the required condition.
• Step 4: Choose the correct answer from the given choices.

Now, let's work through each step:
Step 1: A square root of a number is a value that, when multiplied by itself, gives the original number. Here, we want $y$ such that $y^2 = 36$.
Step 2: We test integer values to find which one squared equals 36. Testing $y = 1, 2, 3, 4, 5,$ and $6$ gives:
- $1^2 = 1$
- $2^2 = 4$
- $3^2 = 9$
- $4^2 = 16$
- $5^2 = 25$
- $6^2 = 36$

Step 3: The integer $6$ satisfies $6^2 = 36$. Therefore, $\sqrt{36} = 6$.

Step 4: The correct choice from the given answer choices is 6 (Choice 4).

Hence, the square root of 36 is $\mathbf{6}$.

The root of 441 is 21.

$21\times21=$

$21\times20+21=$

$420+21=441$

To solve this problem, we'll determine the square root of the number 4.

• Step 1: Recognize that the square root of a number is asking for a value that, when multiplied by itself, yields the original number. Here, we seek a number $y$ such that $y^2 = 4$.
• Step 2: Identify that $4$ is a perfect square. The numbers $2$ and $-2$ both satisfy the equation $2^2 = 4$ and $(-2)^2 = 4$.
• Step 3: We usually consider the principal square root, which is the non-negative version. Thus, $\sqrt{4} = 2$.

Therefore, the solution to the problem is 2, which corresponds to the correct choice from the given options.

To solve this problem, we'll determine the square root of 64, following these steps:

• Step 1: Identify the number whose square root we need to find. The number given is 64.
• Step 2: Determine which number, when multiplied by itself, equals 64.
• Step 3: Recall that $8 \times 8 = 64$.

Now, let's work through each step:

Step 1: We are tasked with finding $\sqrt{64}$. The problem involves identifying the number which, when squared, results in 64.
Step 2: To find this number, we'll check our knowledge of squares. We know that 8 is a significant integer whose square results in 64.
Step 3: Compute: $8 \times 8 = 64$. Hence, $8$ meets the requirement.

We find that the solution to the problem is $\sqrt{64} = 8$.