Square Root of a Negative Number

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Square Root of a Negative Number

There is no root of a negative number since any positive number raised to the second power will result in a positive number.

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\( \sqrt{100}= \)

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Square Root of a Negative Number

Everything you need to know about the root of negative numbers is that... it simply does not exist!
Negative numbers do not have a root, if in an exam you come across an exercise involving the root of a negative number, your answer should be that it has no solution.
Want to understand the logic? Keep reading.


What is the root of a number?

The root is some number, let's suppose one that we will call XX that, in fact, will be positive and that, when multiplied by itself will give us XX.
For example, the root of 100100  will be a positive number that if we multiply it by itself we will obtain 100100 .
That is, 1010.
Instead of saying "multiply it by itself" we can say "raise it to the square".


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Why does a negative number not have a square root?

As we have seen, the root of any number, for example, AA is a positive number that if we square it will give us AA.
There is no positive number in the whole world that when squared will give us a negative number, therefore, negative numbers do not have a root.


Exercise Practice

Solve the exercise:
9=3\sqrt9=3
If we raise 33 to the power of two, we will get 99.
Another exercise
9=No solution\sqrt{-9} = No~solution
We will not be able to find any positive number that, when squared, gives us 9-9 since any positive number squared will be positive and never negative.


Do you know what the answer is?

Examples with solutions for Square Root of a Negative Number

Exercise #1

100= \sqrt{100}=

Video Solution

Step-by-Step Solution

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 x2=100 x^2 = 100 , then x x should be our answer.

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

  • 12=1 1^2 = 1
  • 22=4 2^2 = 4
  • 32=9 3^2 = 9
  • and so forth, up to 102 10^2

Step 2: Checking integers, we find that:

102=10×10=100 10^2 = 10 \times 10 = 100

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

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

Therefore, the square root of 100 is 10.

Answer

10

Exercise #2

16= \sqrt{16}=

Video Solution

Step-by-Step Solution

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: 4×4=16 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 4 .

Answer

4

Exercise #3

25= \sqrt{25}=

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine the square root of 25.

  • Step 1: The square root operation asks us to find a number that, when multiplied by itself, equals the given number, 25.
  • Step 2: Consider what number times itself equals 25. We note that 5×5=255 \times 5 = 25.
  • Step 3: Thus, the square root of 25 is 5.

Therefore, the solution to the problem is 25=5\sqrt{25} = 5.

The correct answer is choice 2: 5.

Answer

5

Exercise #4

36= \sqrt{36}=

Video Solution

Step-by-Step Solution

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 y such that y2=36 y^2 = 36 .
Step 2: We test integer values to find which one squared equals 36. Testing y=1,2,3,4,5, y = 1, 2, 3, 4, 5, and 6 6 gives:
- 12=1 1^2 = 1
- 22=4 2^2 = 4
- 32=9 3^2 = 9
- 42=16 4^2 = 16
- 52=25 5^2 = 25
- 62=36 6^2 = 36

Step 3: The integer 6 6 satisfies 62=36 6^2 = 36 . Therefore, 36=6 \sqrt{36} = 6 .

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

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

Answer

6

Exercise #5

36= \sqrt{36}=

Video Solution

Step-by-Step Solution

Let's solve the problem step by step:

  • Step 1: Understand what the square root means.
  • The square root of a number nn is a value that, when multiplied by itself, equals nn. This is written as x=nx = \sqrt{n}.

  • Step 2: Apply this definition to the number 3636.
  • We are looking for a number xx such that x2=36x^2 = 36. This translates to finding x=36x = \sqrt{36}.

  • Step 3: Determine the correct number.
  • We know that 6×6=366 \times 6 = 36. Therefore, the principal square root of 3636 is 66.

Thus, the solution to the problem is 36=6 \sqrt{36} = 6 .

Among the given choices, the correct one is: Choice 1: 66.

Answer

6

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