What are those mysterious square roots that often confuse students and complicate their lives? The truth is that to understand them, we need to grasp the concept of the inverse operation.
What are those mysterious square roots that often confuse students and complicate their lives? The truth is that to understand them, we need to grasp the concept of the inverse operation.
When we solve an exercise like , it's clear that times (that is, multiplying the number by itself) results in . This is the concept of a power, or to be more precise, a square power, which to apply, we must multiply the figure or the number by itself.
The concept of "square root" refers to the inverse operation of squaring numbers.
That is, if we have and we want to find the value of , what we need to do is perform an identical operation on both sides of the equation.
So, we have: and the result is .
Choose the largest value
Now let's explain in more detail the operation we have performed.
There are two numbers that meet these requirements: and .
With that said, it's important to remember that the square root of a number will always be positive.
Therefore, to summarize, in the exercise: for example, we have two possible answers: and .
If we are given the mathematical expression the only possible answer will be .
You cannot find the square root of a negative number, that is, the expression ย is not correct and has no answer.
On the other hand,
that is, as long as the number under the square root is positive, we can find its square root.
For example:
\( \sqrt{64}= \)
\( \sqrt{36}= \)
\( \sqrt{49}= \)
Simple square roots are based on multiplication tables. Let's see some examples:
Solution: The square root, as we have already seen, is the inverse operation of squaring a number. Therefore, we must ask ourselves which number squared, or which number multiplied by itself, will result in .
Given that:
the answer will be:
Solve the following exercise:
\( \sqrt{x^2}= \)
\( 5+\sqrt{36}-1= \)
\( \sqrt{441}= \)
Solution: in this case, we also have to ask ourselves what number squared, or what number multiplied by itself, results in .
Given that: the answer will be:
Solution: following the same logic, we see that and, therefore, the answer will be: .
\( \sqrt{225}= \)
\( \sqrt{144}= \)
\( \sqrt{121}= \)
In this section, we will apply what we have learned so far regarding square roots and see how we can use this information to solve algebraic exercises that include square roots.
An important rule that we must remember when solving these types of exercises is that:
Square roots should be solved first, that is, before any other mathematical operation that is outside of the square root itself.
Solution:
First, we will solve the square root, as it comes before the addition that is outside of it.
Thus, we obtain:
Then we continue with the rest of the exercise:
Therefore, the result is
\( \sqrt{100}= \)
\( \sqrt{64}= \)
\( \sqrt{36}= \)
Solution:
This exercise is a bit more complicated. Initially, we must solve the square root, as this operation precedes any other in the exercise.
Thus, we obtain:
Afterwards, we should approach the exercise like any other math problem: .
According to the order of operations in math, multiplication and division come before addition and subtraction. Therefore, the result is: .
The answer to the exercise is:
Solution:
Here we also need to solve the square roots first.
We insert the numbers and solve according to the order of operations:
Therefore, the answer is:
Choose the largest value
Let's begin by calculating the numerical value of each of the roots in the given options:
We can determine that:
5>4>3>1 Therefore, the correct answer is option A
Solve the following exercise:
In order to simplify the given expression, we will use two laws of exponents:
a. The definition of root as an exponent:
b. The law of exponents for power of a power:
Let's start with converting the square root to an exponent using the law mentioned in a':
We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:
Therefore, the correct answer is answer a'.
To solve the expression , 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:
Substitute the square root back into the expression:
Next, perform the addition and subtraction from left to right:
Add 5 and 6:
Then subtract 1:
Finally, you obtain the solution:
The root of 441 is 21.
To solve the expression , 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:
Now, substitute back into the expression:
Step 2: Perform the subtraction:
Step 3: Perform the addition:
Therefore, the final answer is .
\( \sqrt{16}= \)
\( \sqrt{9}= \)
Choose the largest value