# Cube & Cubeth Root

Created by: Team Maths - Examples.com, Last Updated: May 7, 2024

## What is Cube?

A cube of a number is the result of multiplying that number by itself three times. It is expressed mathematically as πΒ³Β , where πβΏΒ is any real number. For example, the cube of 4 is 4Β³

## What is Cube Roots?

The cube root of a number is a value that, when cubed, gives the original number. It is denoted as π3. For instance, the cube root of 64 is Β³β64=4,Β because 4Β³ =64

## Properties of Cubes and Cube Roots

Cubes and cube roots possess unique mathematical properties that are essential for both theoretical studies and practical applications. Here are the key properties for each:

## Properties of Cubes

1. Preservation of Sign:

• The cube of a positive number is positive, and the cube of a negative number is negative. This differs from squaring, where both positive and negative numbers yield a positive square.
2. Odd and Even Nature:

• The cube of an odd number remains odd, and the cube of an even number remains even, maintaining the parity of the original number.
3. Sum of Cubes Formula:

• The sum of the cubes of the first πn natural numbers is given by the formula:
1Β³+2Β³+3Β³+β¦+πΒ³=(π(π+1)/2)Β²
• This formula is a square of the sum of the first π natural numbers, illustrating a deep algebraic identity.
4. Multiplicative Property:

• The cube of a product of numbers is equal to the product of their cubes:
(πππ)Β³=πΒ³ΓπΒ³ΓπΒ³

### Properties of Cube Roots

1. Inverse Operation:

• The cube root is the inverse operation of cubing a number. If π₯Β³=π¦, then Β³βy=x.
2. Behavior with Negative Numbers:

• Unlike square roots, cube roots can be taken of negative numbers. For example, Β³ββ8=β2
3. Distribution over Multiplication:

• Cube roots can be distributed over multiplication:
Β³βπππ=Β³βπΓΒ³βπΓΒ³βπ
4. Root of a Fraction:

• The cube root of a fraction is the fraction of the cube roots:
Β³βπ/b = Β³βπ/Β³βπβ
5. Continuity:

• Cube root functions are continuous for all real numbers, meaning they do not have any breaks or jumps in their graphs.

## Cube Root Formula

The cube root of a number π provides a value π such that when π is cubed, it returns π. Mathematically, this relationship is expressed as: Β³βπ=πThis equation holds true under the condition that: π=πΒ³This relationship is particularly useful when determining the cube root of perfect cubes, where π and π

## How to Find Cube Root of a Number?

### Step 1: Prime Factorization

Begin by breaking down the number into its prime factors. This involves dividing the number by the smallest prime number (starting with 2) and continuing the process with the quotient until only prime numbers are left.

### Step 2: Group the Factors

Organize the resulting prime factors into groups, each containing three identical factors. This step is crucial because the cube root of a number will be the product of factors from each group.

### Step 3: Calculate the Cube Root

Multiply one factor from each group to find the cube root. If any factors remain that cannot be grouped into threes, then the number is not a perfect cube, and an exact cube root (in whole numbers) cannot be determined.

### Question 1: Solve the Equation

Solve the equation 2π₯β6=16

Answer: Add 6 to both sides: 2π₯=22 Divide by 2: π₯=11Final Answer: π₯=11

### Question 2: Calculate Compound Interest

Calculate the compound interest on $1000 for 2 years at an annual interest rate of 5%, compounded annually. Answer: Use the formula for compound interest: π΄=π(1+π)πA=P(1+r)n Where: • π=1000 • π=0.05 • π=2 Calculate: π΄=1000(1+0.05)Β² π΄=1000(1.05)Β² π΄=1000Γ1.1025=1102.5 Final Answer: The compound interest is$1102.5.

### Question 3: Convert Temperatures

Convert 68Β°F to Celsius using the formula πΆ=5/9(πΉβ32)C=95β(Fβ32).

Answer: Plug in the Fahrenheit value: πΆ=5/9(68β32)

πΆ=5/9(36)

πΆ=20

## What is Cube or Cubed Root?

A cube is a number raised to the power of three, expressed as πΒ³. A cube root is the inverse, finding a number πn that when cubed gives the original number, denoted as Β³βΓ

## What is the Cube Root Formula?

The cube root formula to find πn from πΒ³=π₯ is expressed as Β³βΓ = nIt calculates the number π whose cube equals π₯.

## What is the 729 Cube Root?

The cube root of 729 is 9, since 9Β³=729 This means Β³β729=9.

## What is a Cube of 4?

The cube of 4 is 64, calculated as 4Β³=4Γ4Γ4=64

## What is Cube 1 to 10?

Cubes from 1 to 10 are: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. These represent πΒ³ for πn from 1 to 10.

## Is 30 a Cube Number?

No, 30 is not a cube number. There is no integer πn such that πΒ³=30. The cube numbers closest to 30 are 27 (3Β³) and 64 (4Β³)

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