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

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'.

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 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 $150$.

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 $\boxed{18.8}$.