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Square roots and cube roots

Square roots and cube roots

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Proportion


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Summary

Square roots and cube roots

In a nutshell

Square roots and cube roots are the inverses of the square and cube operations.



Square roots

The square root of a number, xx, is denoted as x\sqrt{x}​. A square root is defined as follows:

If x=y2x=y^2​, then x=y\sqrt{x}=y​.


In other words, finding the square root of xx​ is the same thing as asking what number becomes xx when multiplied by itself?


Key facts to know about square roots

  • The square root of a number is always defined to be positive.
  • The square root of a negative number does not exist.


Finding square roots

Square roots can either by found by memorising square numbers, or by using a calculator.


Example 1

What is the square root of 3636?


You should know that 62=6×6=366^2=6\times6=36. Therefore, 36=6\underline{\sqrt{36}=6}.


Example 2

What is the square root of 8585?


Use the \sqrt{} button on your calculator:

85=9.219544457...\underline{\sqrt{85}=9.219544457...}​​



Cube roots

The cube root of a number, xx​, is denoted as x3\sqrt[3]{x}​. A cube root is defined as follows:

If x=y3x=y^3​, then x3=y\sqrt[3]{x}=y.


In other words, finding the cube root of xx is the same as asking what number, when cubed, becomes xx?


Key facts to know about cube roots

  • You can cube root negative numbers.
  • The cube root of a negative number is negative.


Finding cube roots

Cube numbers can be very large for even smaller numbers. So, in most cases, you will have to use your calculator to find cube roots.


Example 3

What is the cube root of 512512?


83=8×8×8=5128^3=8\times8\times8=512, so 5123=8\underline{\sqrt[3]{512}=8}.


Example 4

What is the cube root of 100-100?


Use the 3\sqrt[3]{} button on your calculator:

1003=4.641588834...\underline{\sqrt[3]{-100}=-4.641588834...}​​


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Exercises

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FAQs - Frequently Asked Questions

How do you find the square root or cube root of a number?

What is the cube root of a number?

What is the square root of a number?

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