How to Calculate the Cube Root of 343: Step-by-Step Guide
Cube root of 343, represented as 3√343, is a value that returns the original value when multiplied three times by itself. This is the most fundamental definition of a cube root. If an integer is a perfect cube, we can calculate its cube root. We shall get the value of the cube root of 343 in this article by utilizing the prime factorization technique and examples with detailed explanations.
What is the Cube Root?
When a number (x) is multiplied three times, the resulting number is called its cube. As a result, the cube of (x) becomes x3 or x-cubed.
For Example: If we multiply 9 three times, we get 729.
9 x 9 x 9 = 729.. Thus, 9 will be referred to as the cube root of 729.
Symbol of a Cube Root
The symbol that we use to represent a cube root is the same as that of a square root with the only difference that in a square root, we use the number 2 and in cube root, we use the number 3. The root symbol can also be called a radical symbol. Here is how we represent a cube root:
∛x
What is the Cube Root of 343?
The cube root of 343 is the integer that, when multiplied three times by itself, produces the result 343. Because 343 can be written as 7 7 7. As a result, the cube root of 343 is ∛(7 × 7 × 7) = 7.
Ways to find Cube root
Typically, we apply the prime factorization method to discover the prime factors of a given number. Therefore, if we examine 343 prime factors, we must organize them into groups of three, which provides the cube of factors. Thus, once we have determined the cube of a number's factors, we can apply the cube root, which cancels out the cubes.
Step-by-Step Procedure to find a Prime Factor of 343
Step 1:
Determine the prime factors of 343.
343 equals 7 x 7 x 7
Step 2:
It is self-evident that 343 is a perfect cube. As a result, arrange the 343 components into three pairs and write them as cubes.
343 equals 7x7x7
343 = 73
Step 3:
We will apply the cube root to both sides of the preceding statement to eliminate the word in cubes.
3√343 = 3√(73)
Thus, the cube root is canceled out in this case by the cube of 7.
As a result, 3√343 = 7
As a result, the value of 343's cube root is 7.
Examples of Cube Root of 343
Example 1: What is the cube root of 15625?
Solution:
5x5x5x5x5x5 is the cube root of 15625.
= (5x5)x(5x5)x(5x5)
=25x25x25
As a result, 25 equals the cube root of 1562
Example 2: Simplify this expression: 2× 3√343+ 10?
Solution:
We know, 3√343 = 7.
Now, substitute this into the expression.
2×3√343 + 10 = 2(7) + 10
2×3√343 + 10 = 14+10 = 24.
=2×3√343 + 10 = 24.
Example 3: What can the smallest integer be divided by 73002 to form a perfect cube?
Solution:
73002 has a prime factorization of 23 x 23 x 23 x 2 x 3.
There is already a set of three 23s here, but only 2 and 3 remain. Thus, by dividing the number by 6, we may obtain a perfect cube.
FAQs on Cube Root of 343 Explained with Methods and Examples
1. What is the value of the cube root of 343?
The cube root of 343 is 7. This is because when 7 is multiplied by itself three times, it equals 343 (7 x 7 x 7 = 343). The mathematical representation is ∛343 = 7.
2. How can you find the cube root of 343 using the prime factorisation method?
To find the cube root of 343 using the prime factorisation method, you follow these steps:
Step 1: Find the prime factors of the number 343. The only prime factor is 7, so 343 = 7 × 7 × 7.
Step 2: Group the identical factors in triplets. Here, we have one group of three 7s: (7 × 7 × 7).
Step 3: Take one factor from each triplet. From the group, we take a single 7.
Therefore, the cube root of 343 is 7.
3. Is 343 considered a perfect cube? Explain why.
Yes, 343 is a perfect cube. A number is classified as a perfect cube if it can be obtained by multiplying an integer by itself three times. Since 343 is the result of 7 × 7 × 7 (or 7³), it perfectly fits this definition.
4. What is the difference between the square root of 343 and the cube root of 343?
The primary difference between the square root and the cube root is the power involved:
The cube root of 343 is 7, as 7³ = 343.
The square root of 343 is an irrational number, approximately 18.52, as 18.52² ≈ 343.
In essence, the cube root finds a number that gives 343 when cubed, while the square root finds a number that gives 343 when squared.
5. If a cube-shaped box has a volume of 343 cm³, what is the length of one of its sides?
This is a real-world application of cube roots. The volume of a cube is given by the formula V = side³. To find the side length, you must calculate the cube root of the volume.
Given Volume (V) = 343 cm³
Side = ∛V = ∛343
Since ∛343 = 7, the length of one side of the box is 7 cm.
6. How does knowing the cube root of 343 help in finding the cube root of 0.343?
Knowing that ∛343 = 7 simplifies calculating the cube root of a decimal like 0.343. You can express 0.343 as a fraction, 343/1000. Using the cube root property ∛(a/b) = (∛a) / (∛b):
∛(0.343) = ∛(343 / 1000)
= (∛343) / (∛1000)
= 7 / 10
= 0.7
Thus, the cube root of 0.343 is 0.7.
7. Can you use the estimation method to find the cube root of 343? Explain the steps.
Yes, the estimation method works well for perfect cubes like 343. Here’s how:
Step 1: Observe the digit in the units place of 343, which is 3. In cubes and cube roots, a number ending in 3 will have a cube root ending in 7 (since 7³ = 343).
Step 2: Since 343 is a three-digit number that is less than 1000 (10³), its cube root must be a single-digit number.
Combining both steps, the only single-digit number that ends in 7 is 7 itself. Therefore, the estimated cube root is 7.
8. Why does the positive number 343 have only one real cube root and not two?
A positive number has only one real cube root due to the properties of multiplying negative numbers. While a square root can be positive or negative (e.g., 5²=25 and (-5)²=25), this is not true for cube roots.
A positive number cubed is positive: 7 × 7 × 7 = 343.
A negative number cubed is negative: (-7) × (-7) × (-7) = -343.
Because a negative number cannot be cubed to result in a positive number, 343 has only one real cube root, which is 7.
