Cube and cube roots

Learning Outcomes

After reading the post readers will be able to,

• Find the cube of a number.

• Know about the perfect cubes.

• Know about the cube root and how to find the cube root of a giving number.

Cubes:

Cube of a number means to multiply a number by itself three times. Suppose y be any number then, 

y×y×y = y³

For example

2×2×2 = 2³

3×3×3 = 3³

4×4×4 = 4³

5×5×5 = 5³

6×6×6 = 6³

7×7×7 = 7³

And so on.

Perfect cubes:

Perfect cube is a number that is the result of multiplying (product) an integer by itself three(3) times. It can also be define as an integer to the third power of another Integer.

Example: show that 8 is a perfect cube.

Solution: 8 = 2×2×2 = 2³ so 8 is a perfect cube of 2.

Example: show that 27 is a perfect cube.

Solution: 27 = 3×3×3 = 3³

So 27 is a perfect cube of 3.

Example:  Is 216 a perfect cube?

Solution:

216 = 2×2×2×3×3×3 = 2³ × 3³

= (2×3)³ = 6³ so 216 is a perfect cube of 6.

Example: show that 343 is a perfect cube.

Solution:

343 = 7×7×7 = 7³ so 343 is a perfect cube of 7.

Example: find the cube of 5.

Solution: (5)³ = 5×5×5 = 125 Ans.

Example: find the cube of 3.

Solution: (3)³ = 3×3×3 = 27 Ans.

Example: find the cube of (1.2).

Solution: (1.2)³ = (1.2)×(1.2)×(1.2) = (1.44)×(1.2) = 1.728 Ans.

Cube Roots:

In math a cube root of a number is denoted by y⅓ is a number such that a³ = y i.e a = y⅓.

The symbol of cube root is ∛.

Example: find the cube root of 125.

Solution: Factorization of 125 is

125 = 5×5×5 = 5³

∛(125) = ∛(5×5×5) = (5³)⅓ = 5 Ans.

Example: find cube root of 9261.

Solution: Factorization of 9261 is

9261 = 3×3×3×7×7×7 = 3³ × 7³

∛(9261) = ∛(3³×7³) = (3³×7³)⅓ = (3³)⅓ × (7³)⅓ = 3×7 = 21

Properties of cubes of numbers:

1) Cube of a positive (+) number is positive (+). For example 3³ = 27

2) Cube of a negative (-) number is negative (-). For example (-4)³ = -64

3) Cube of an even numbers is even. For example 6³ = 216

4) Cubes of  odd numbers is odd. For example 7³ = 343

5) Cube of distributive property under multiplication is

(5×8)³ = 5³ × 8³

and cube of distributive property under division is

(5/8)³ = 5³/8³

6) Cube number of the perfect cubes

2³ = 8 ,

6³ = 216 ,

8³ = 512 so 8, 216 and 512 are perfect cubes.

Questions:

Q1 : Is 512 perfect cube?

ANS: first we factorize 512 as

512 = 2×2×2×2×2×2×2×2×2

512 = 2³ × 2³ × 2³

512 = 8×8×8

512 = 8³

So 512 is a perfect cube.

Q2:  81 is a perfect cube ?

ANS: first we factorize 27 as

81 = 3 ×3×3×3

81 = 3³ × 3¹

So 81 is not a perfect cube.

Question 3: is 6859 perfect cube?

Answer: first we factorize it as

6859 = 19×19×19

6859 = 19³

So 6859 is a perfect cube.

Question 4: 729 is a perfect cube?

Answer: first we factorize it as

729 = 9×9×9

729 = 9³

So 729 is a perfect cube.

Question 5: (1000/125) is a perfect cube?

Answer: first we factorize it as

(1000/125) = (10×10×10)/(5×5×5)

(1000/125) = 10³/5³

(1000/125) = (10/5)³

So (1000/125) is a perfect cube.

Example: find the cube of (1.4)

Solution:

Taking cube of (1.4)

(1.4)³ = (1.4) × (1.4) × (1.4)

(1.4)³ = (1.96) × (1.4)

(1.4)³ = 2.744 Ans.

Example: find cube of (0.8)

Solution: Taking cube of (0.8)

(0.8)³ = (0.8) × (0.8) × (0.8)

(0.8)³ = (0.64) × (0.8)

(0.8)³ = 0.512 Ans.

Example: find the cube root of 729.

Solution:

Taking cube root of 729.

∛729 = ∛(3×3×3×3×3×3)

∛(729) = ∛(3³) × ∛(3³)

∛(729) = (3³)⅓ × (3³)⅓

∛(729) = 3×3 = 9 Ans.

Example: find the cube root of 15625.

Solution:

Taking cube root of 15625.

∛(15625) = ∛(5×5×5×5×5×5)

∛(15625) = ∛(5×5×5) × ∛(5×5×5)

∛(15625) = ∛(5³) × ∛(5³)

∛(15625) = (5³)⅓ × (5³)⅓

∛(15625) = 5×5 = 25 Ans.

Example: Find the cube root of 13824.

Solution:

Taking cube root of 13824.

∛(13824) = ∛(2×2×2×2×2×2×2×2×2×3×3×3)

∛(13824) = ∛(2³) × ∛(2³) × ∛(2³) × ∛(3³)

∛(13824) = (2³)⅓ × (2³)⅓ × (2³)⅓ × (3³)⅓

∛(13824) = 2×2×2×3

∛(13824) = 24 Ans.

Frequently asked questions. (FAQs)

Are cubes and cube roots same?

we understand this by a example, 216 is called the cube of 6. While on the other hand, the cube root of a number is the reverse or opposite process of the cube of a number. The cube root of a number is written as ∛ . Considering the same example, 2 is called the cube root of the number 8.

What is the cube of √ 3?

The cube root of 27 is 3 and of 8 is 2. Cube root of any number y is a number x, such as x³= y. So, to find the cube root of 3 we have to determine a number, which when multiplied three times, gives the number 3, such as x³ = 3 or x = ³√3.

What is Cube Root of a Number?

The cube of a number is the value of the third exponent or power three of the number. For example, the cube of 3 is 3³= 3 × 3 × 3 = 27. While the cube root is the reverse process or opposite process of the cube of a number and is denoted by ∛. For example, ∛27, that is, the cube root of 27 = 3 because when 3 is multiplied three time with itself, it gives 27.

What is Cube Root Used For?

The cube root is used to solve the cubic equations. Cube roots are also used to find the dimensions of a cube if its volume is given.

How do you find cube root of a number? 

We define the cube root as, "Cube root is the number which needs to be multiplied thrice or multiplied three times to get the original number." Now, let us look at the cube root formula, where z is the cube root of x. ∛x = z.

How we find the cube root of 64?

The cube root of 64 is the number which when multiplied by itself three times which gives the product as 64. Since 64 can be expressed as 2 × 2 × 2 × 2× 2× 2. So, the cube root of 64 = ∛(2 × 2 × 2 × 2 × 2 × 2) = 4.

What is a example of cube root ?

The cube root of number is a value when we multiplied by itself three times or thrice produces the original value. For example, the cube root of 64, denoted as ³√64, is 4, because when we multiply 4 by itself three times we get 4 x 4 x 4 = 64 = 4³ or 2⁶

Find the cube root of 4?

The cube root of 4 is a number, when we multiplied by itself three times or thrice it gives the product as 4. So 4 can be expressed as 2×2. The cube root of 4 = ∛(2 × 2) = 1.587.

What is the fast way to find cubes of given numbers?

Cube of a number means, a number that is multiplied by itself thrice or three times, then the product is called the cube of that given number.

1³=1×1×1=1

2³=2×2×2=8

3³=3×3×3= 27 and so on..

4³= 64 

6³= 216 

7³= 343 

8³= 512 

9³= 729 

10³= 1000

What does cube root look like?

The cube root ( ∛ ) of a number is written with a small number 3 which is called the index, just outside and above the radical symbol.

This small or little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol.

What is the cube root symbol?

The cube root of a number is the factor that we multiply by itself three times or thrice to get that number. The symbol for cube root is ∛ .

Have a nice day.