Square Root of a Negative Number

๐Ÿ†Practice square roots

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.

Start practice

Test yourself on square roots!

einstein

Choose the largest value

Practice more now

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 XXย that, in fact, will be positive and that, when multiplied by itself will give us XX.
For example, the root of 100100 ย 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".


Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge

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
โˆ’9=Noย solution\sqrt{-9} = No~solution
We will not be able to find any positive number that, when squared, gives us โˆ’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

Choose the largest value

Video Solution

Step-by-Step Solution

Let's begin by calculating the numerical value of each of the roots in the given options:

25=516=49=3 \sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\ We can determine that:

5>4>3>1 Therefore, the correct answer is option A

Answer

25 \sqrt{25}

Exercise #2

64= \sqrt{64}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Recognize what finding a square root means
  • Step 2: List known perfect squares to identify which one results in 64
  • Step 3: Verify the square root by calculation

Now, let's work through each step:
Step 1: To find the square root of 64, we seek a number that, when multiplied by itself, equals 64.
Step 2: Consider the sequence of perfect squares: 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 , 72=49 7^2 = 49 , 82=64 8^2 = 64 .
Step 3: We see that 82=64 8^2 = 64 . Therefore, the square root of 64 is 8.

Therefore, the solution to this problem is 8 8 .

Answer

8

Exercise #3

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 6ร—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

Exercise #4

49= \sqrt{49}=

Video Solution

Step-by-Step Solution

To solve this problem, we follow these steps:

  • Step 1: Understand that finding the square root of a number means determining what number, when multiplied by itself, equals the original number.
  • Step 2: Identify the numbers that could potentially be the square root of 4949. These are ยฑ7 \pm7, but by convention, the square root function typically refers to the non-negative root.
  • Step 3: Calculate 7ร—7=497 \times 7 = 49. This confirms that 49=7 \sqrt{49} = 7.
  • Step 4: Verify using the problem's multiple-choice answers to ensure 77 is among them, confirming choice number .

Therefore, the solution to the problem 49 \sqrt{49} is 7 7 .

Answer

7

Exercise #5

Solve the following exercise:

x2= \sqrt{x^2}=

Video Solution

Step-by-Step Solution

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}} b. The law of exponents for power of a power:

(am)n=amโ‹…n (a^m)^n=a^{m\cdot n}

Let's start with converting the square root to an exponent using the law mentioned in a':

x2=โ†“(x2)12= \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:

(x2)12=x2โ‹…12x1=x (x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x} Therefore, the correct answer is answer a'.

Answer

x x

Start practice