How to Find the Cube Root of 729

Cube Root of 729

Have you ever looked at a big number and thought, "How can I break it down into its smallest parts?" For a lot of students, the hard part isn't just knowing the answer; it's also figuring out how to get there. It's important to know how to get the cube root of 729, whether you're working on a geometry issue that involves volume or just trying to get better at mental arithmetic. This method is the opposite of "cubing" a number. To determine the cube root of a number, we have to start with the total and work our way back. For example, to obtain the cube root of 9, we would have to do 9 × 9 × 9. This article will show you the easiest ways to figure this out so that you never get stuck on this maths problem again.

The cube root of a number is a number that, when multiplied by itself three times, equals the original number. The symbol ∛ stands for the cube root.

So, when we have to find the cube root of 729, we are looking for a value (let's call it 'x') such that:

• x × x × x = 729

• x³ = 729

As we will discover through our calculations, the cube root of 729 is exactly 9. This is because 9 multiplied by 9 is 81, and 81 multiplied by 9 results in 729.

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Steps to Find Cube Root of 729

There are different ways to find the cube root of 729, but the most reliable and commonly used technique in school exams is the prime factorisation method.

1. The Prime Factorisation Method

You split 729 down into prime factors, which are numbers that can only be divided by 1 and themselves.

Steps for finding the cube root of 729:

• Step 1: Divide 729 by the smallest possible prime number. Since 729 is odd, we try 3.

• Step 2: 729 ÷ 3 = 243

• Step 3: 81 is the answer to 243 ÷ 3.

• Step 4: 27 ÷ 3 = 81

• Step 5: 3 times 27 equals 9.

• Step 6: 3 ÷ 9 = 3

• Step 7: One is the answer to 3 ÷ 3.

The prime factors of 729 are 3 × 3 × 3 × 3 × 3 × 3.

To find the cube root, we group these factors into sets of three (triplets):

• (3 × 3 × 3) × (3 × 3 × 3)

From each triplet, we take one number:

• 3 × 3 = 9

Therefore, ∛729 = 9.

2. The Estimation Method

If you know common cubes, you can spot the answer quickly.

• We know that 8³ = 512

• We know that 10³ = 1000

• The cube root must be between 8 and 10 because 729 is between 512 and 1000. Also, numerals that end in 9 generally make cubes that end in 9. The cube root is 9 because 9 × 9 × 9 finishes at 9 and delivers 729 precisely.

Read More - Perfect Cube: Definition, Formula, List of Perfect Cubes 1 to 50

Differences Between Square Roots and Cube Roots of 729

Many learners get confused between cube root and square root. Always remember that cube roots use groups of three identical factors, whereas, square roots use groups of two. You may also be asked to find the square root of 729 by prime factorization method to compare both ideas side by side.

Finding the Square Root of 729 via Prime Factorisation

To determine the square root of 729, we use the same factors but group them in pairs (twos) instead of triplets (threes):

• Prime factors: 3 × 3 × 3 × 3 × 3 × 3

• Pairs: (3 × 3) × (3 × 3) × (3 × 3)

• Take one from each pair: 3 × 3 × 3 = 27

When you split 729 down into prime factors, you get 9 as the cube root and 27 as the square root.

Read More - (A - B)^3 Formula | A Minus B Whole Cube Formula with Examples

Cube Root of 729 Example Problems

Example 1: Finding the side of a cube

What is the length of one side of a cubic box that holds 729 cubic centimetres?

Solution: The volume of a cube is equal to the side cubed. So, 

• 729 = Side³

• Side = ∛729

• Side = 9 cm.

Example 2: Simple Equation

Solve for 'y' if y³ - 729 = 0.

• Solution: y³ = 729

• y = ∛729

• y = 9.

Summary Table for 729
Let’s summarise the cube root and square root of 729 for revision.

 Once you know how to do prime factorisation, figures like 729 will be easy for you to work with. You may use this method on any perfect cube, and it will always work. It will help you develop a strong base for more advanced algebra and mental maths.

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