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

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:

$10$

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 $81+\sqrt{81}+10$, 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:

$100$