4.2 Cubic and cube roots

It was said that he radical sign was first used by Rene Descartes, but long before he used this, it has been developed in Germany in the 1500s. The radical ( sign was invented to find the square root or the cube root, 4th root, 5th root etc., depending on the index of the radical sign.

Now if you remember, in previous chapter, we learned all about the different number systems and we have mentioned about the rational and irrational numbers. Radicals can belong to either of which depending of course if the radicals would be able to fit into the definition of whichever group of number. In this lesson we will get to be able to identify whether or not a certain radical is rational. We will be specifically going to study about a square root or a cube root of a particular number.

We will first look into what a square and a cube are since these would be the two fundamental concepts. In order to solve a square root, or a cube root, one must know how to square or cube a number. These will be thoroughly discussed in 4.1 and 4.2.

For 4.3 to 4.8 we will be looking closer into how to simplify and evaluate these radicals through applying what we have learned from chapters on square and square roots, estimating square roots, and rational numbers from earlier grades. We would also be learning how to convert them into either entire radicals to mixed radicals or vice versa, by applying our knowledge of factoring out the perfect squares or the perfect cubes in the equation. Apart from converting the radicals from one form to another we are also going to learn more about how to combine different radicals and simplify them depending on the operation used and learn to rationalize a certain radical equation.

Cubic and cube roots

Whenever we see “roots”, let it be cubic roots or square roots, we know for sure that we will need to do prime factorization to find out the prime factors of the numbers. In this section, we use factors and multiples to find perfect cube whole numbers and cubic roots.


To cube:
Raise the number to the third power
Ex: 33 {3^3} =3×3×3=27= 3\times 3\times 3 = 27
63 {6^3} = 6×6×6=2166\times 6\times 6 = 216

To cube root:
Finding the three identical factors
Ex: 364 {^3}\sqrt{64} = 34×4×4{^3}\sqrt{4\times 4\times 4} = 4
3125 {^3}\sqrt{125} = 35×5×5{^3}\sqrt{5\times 5\times 5} = 5

Perfect Cubes:
03 {0^3}= 0
13 {1^3} = 1
23 {2^3} = 8
33 {3^3} = 27
43 {4^3} = 64
53 {5^3} = 125
63 {6^3} = 216
73 {7^3} = 343
83 {8^3} = 512
93 {9^3} = 729
103 {10^3} = 1000
  • 1.
    Understanding the negative cube roots of the following
  • 2.
    Find the cube roots
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Cubic and cube roots

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