CUBES AND CUBE ROOTS

Think of squaring a number - multiplying it by itself twice. Cube takes that concept further, multiplying the number by itself three times to unveil its "cubed" form. But what if you could rewind this process, finding the original number from its transformed "cube"? That's where the cube root comes in, acting as the key to unlock the original value hidden within the cubed one. Both operations are interconnected, offering different lenses to explore the depths of numbers

What are Cubes and Cube roots?

Cubes:

Imagine raising a number to the power of 3. This simply means multiplying the number by itself three times. The result is called the cube of the original number. For example, the cube of 2 is 2 cubed, which is 2 x 2 x 2 = 8.
Cubes often come up in real-world applications like calculating volumes (think of a cube-shaped box) or growth rates (imagine tripling your investment in three years).

Cube Roots:

Now, flip the script. The cube root of a number is the number you would need to multiply by itself three times to get the original number. It's essentially the opposite of cubing. For example, the cube root of 8 is 2, because 2 x 2 x 2 = 8.
Finding cube roots can be useful for things like understanding geometric shapes or calculating certain scientific variables.

The Relationship:

Think of cubes and cube roots as two sides of the same coin. Cubing is like building a tower with three identical number blocks, while finding the cube root is like deconstructing that tower back into its single block.
They are inverse operations, meaning performing one after the other will bring you back to the original number. For example, if you cube 3 (3 x 3 x 3 = 27), then take the cube root of 27 (3√27 = 3), you get back to 3.

What are Perfect Cubes?

Perfect cubes are numbers that can be expressed as the cube of an integer. In other words, a perfect cube is a number that results from multiplying an integer by itself twice. Mathematically, if $n$ is an integer, then $n^{3}$ is a perfect cube.

For example:

$2^{3} = 2 \times 2 \times 2 = 8$
$(- 3)^{3} = (- 3) \times (- 3) \times (- 3) = - 27$
$4^{3} = 4 \times 4 \times 4 = 64$

Here are the first few perfect cubes:

$1^{3} = 1, 2^{3} = 8, 3^{3} = 27, 4^{3} = 64, 5^{3} = 125, \dots$

Perfect cubes have some interesting properties:

The cube of any integer is an integer.
The cube of a positive number is positive, and the cube of a negative number is negative.
The sum or difference of two perfect cubes can be factored using special formulas, such as $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ and $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$

Cube of Negative Numbers

When you cube a negative number, you are raising that number to the power of 3. The key point to remember is that the cube of a negative number is negative.

For example:

In general, if $n$ is a negative number, then $(- n)^{3}$ will be negative, and the result is obtained by multiplying $n$ by itself twice. The negative sign is retained in the final result.

Cube of 1 to 30 Table

Number	Cube
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000
11	1331
12	1728
13	2197
14	2744
15	3375
16	4096
17	4913
18	5832
19	6859
20	8000
21	9261
22	10648
23	12167
24	13824
25	15625
26	17576
27	19683
28	21952
29	24389
30	27000

How to Find Cube Root through Prime Factorisation?

Finding the cube root of a number using prime factorization involves breaking down the number into its prime factors and then extracting the cube root from those factors. Here's a step-by-step guide:

Step 1: Prime Factorization

Express the given number as the product of its prime factors.

Step 2: Group the Prime Factors

Group the prime factors into sets of three, as each set represents a cube.

Step 3: Identify the Cube Roots

Identify the cube roots of each group.

Step 4: Multiply the Cube Roots

Multiply the cube roots obtained in the previous step to get the final result.

Let's go through an example:

Example: Find the cube root of 216 using prime factorization.

Step 1: Prime Factorization of 216

Step 2: Group the Prime Factors

Group the prime factors into sets of three:

Step 3: Identify the Cube Roots

The cube root of is 2.
The cube root of is 3.

Step 4: Multiply the Cube Roots

Multiply the cube roots obtained:

Therefore, the cube root of 216 is 6.

Solved Examples of Cube and Cube Root

Cube Examples:

Cube of 5: 5 * 5 * 5 = 125
Cube of -2: (-2) * (-2) * (-2) = -8
Cube of 1.5: (1.5) * (1.5) * (1.5) = 3.375

Cube Root Examples:

Cube Root of 8: ∛8 = 2 (since 2 * 2 * 2 = 8)
Cube Root of 64: ∛64 = 4 (since 4 * 4 * 4 = 64)
Cube Root of 125: ∛125 = 5 (since 5 * 5 * 5 = 125)

Examples using Prime Factorization:

Cube Root of 27:
- Prime Factorize: 27 = 3 * 3 * 3
- Group factors: (3 * 3 * 3)
- Take cube root of each group: 3
- Therefore, ∛27 = 3
Cube Root of 648:
- Prime Factorize: 648 = 2 * 2 * 2 * 2 * 2 * 3 * 3 * 7
- Group factors: (2 * 2 * 2), (3 * 3 * 3)
- Take cube root of each group: 2, 3
- Multiply: 2 * 3 = 6
- Therefore, ∛648 = 6

Examples using Column Method:

Cube Root of 729:
- Estimate: Closest perfect cube is 64 (4^3).
- Square estimate: 4^2 = 16.
- Subtract and bring down: 729 - 16 = 713.
- Divide and guess: 713 / (3 * 4) = 59 (rounded down).
- Square and multiply: 59^2 * 2 * 4 = 4504.
- Subtract and compare: 713 - 4504 = -3791 (negative).
- Adjust guess: Increase by 1 (new guess = 60).
- Repeat steps until accuracy is reached.
- Therefore, ∛729 ≈ 9.02 (approximate due to non-perfect cube).
Cube Root of 216:
- Estimate: Closest perfect cube is 125 (5^3).
- Square estimate: 5^2 = 25.
- Subtract and bring down: 216 - 25 = 191.
- Divide and guess: 191 / (3 * 5) = 12 (rounded down).
- Square and multiply: 12^2 * 2 * 5 = 1440.
- Subtract and compare: 191 - 1440 = -1249 (negative).
- Adjust guess: Increase by 1 (new guess = 13).
- Repeat steps until accuracy is reached.
- Therefore, ∛216 = 6.

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CUBES AND CUBE ROOTS

CUBES AND CUBE ROOTS

Cube of 1 to 30 Table

Solved Examples of Cube and Cube Root

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