Square root and cube root question

In summary, the conversation discusses the difference between taking the square root and cube root of a negative number. While it is possible to take the square root of a negative number, the result will be an imaginary number. However, the cube root of a negative number will always result in a real number, regardless of the sign. This idea can also be extended to odd and even roots.
  • #1
Mejiera
15
0

Homework Statement




why is it possible to take the cube root of a negative number and not a square root of a negative number?

Homework Equations





The Attempt at a Solution

 
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  • #2
It is possible to take the square root of a negative number, but you don't get a real number.

If you can write the number whose square root you want as the product of two equal factors, the square root of that number will be one of those factors. For example, 4 = 2*2, so the square root of 4 is 2. Every positive number has two square roots: a positive square root and a negative square root. The symbol [itex]\sqrt{n}[/itex] is taken to mean the principal or positive square root.

The cube root of a number (positive or negative) is one of three equal factors of that number, so [tex]\sqrt[3]{27} = 3[/tex], since 27 = 3*3*3. Similarly, [tex]\sqrt[3]{-8} = -2[/tex] since -8 = (-2)(-2)(-2). Real numbers have only 1 real cube root.

HTH
 
  • #3
Make an assume that it is possible , and work in reverse.. you will realize something tat contradicts from what you've learnt.
 
  • #4
oh I see so for a cube root you a real number answer no matter if its negative or positive ,but for a square root you get an imaginary number if its negative. Mark and icy thanks now I understand.
 
  • #5
And the same idea can be extended to odd and even roots. An even root (square root, fourth root, sixth root, etc.) of a nonnegative real number gives you a nonnegative real number. An odd root (cube root, fifth root, etc.) of a real number gives you a real number with the same sign.
 

1. What is a square root and cube root?

A square root is a number that, when multiplied by itself, gives the original number. It is denoted by the symbol √. A cube root is a number that, when multiplied by itself three times, gives the original number. It is denoted by the symbol 3√.

2. How do you find the square root and cube root of a number?

To find the square root of a number, you can use a calculator or use the long division method. To find the cube root of a number, you can use a calculator or estimate using the prime factorization method.

3. What is the difference between a square root and a cube root?

The main difference between a square root and a cube root is the number of times the number is multiplied by itself. A square root is multiplied by itself once, while a cube root is multiplied by itself three times.

4. Can you find the square root or cube root of a negative number?

No, it is not possible to find the square root or cube root of a negative number. This is because when a negative number is multiplied by itself, it results in a positive number. Therefore, there is no number that when multiplied by itself can give a negative number as the result.

5. What is the significance of square root and cube root in mathematics?

Square root and cube root are important concepts in mathematics as they help in finding the side lengths of squares and cubes, respectively. They are also used in various mathematical equations and in real-life applications such as engineering, science, and finance.

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