Square Root of a Negative Number

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'.

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 root of 441 is 21.

$21\times21=$

$21\times20+21=$

$420+21=441$

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 $150$.