How to Find the Cube Root of 343 Using Prime Factorization
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 What Is the Cube Root of 343
1. What is the cube root of 343?
The cube root of 343 is 7. This is because 7 × 7 × 7 = 343, which means 7 multiplied by itself three times gives 343. In exponential form, this can be written as 7³ = 343, so ∛343 = 7.
2. How do you find the cube root of 343 step by step?
The cube root of 343 is found by prime factorization, which gives the final answer 7.
- Step 1: Prime factorize 343 → 343 = 7 × 7 × 7
- Step 2: Group the factors in sets of three identical numbers.
- Step 3: Take one number from the group → 7
3. Why is 343 a perfect cube?
343 is a perfect cube because it can be written as 7³. A perfect cube is a number obtained by multiplying a whole number by itself three times. Since 7 × 7 × 7 = 343, 343 satisfies the definition of a perfect cube.
4. What is the formula for finding a cube root?
The formula for cube root is ∛a = a1/3. This means the cube root of a number is the value which, when multiplied by itself three times, gives the original number. For example, ∛343 = 3431/3 = 7 because 7³ = 343.
5. Is the cube root of 343 rational or irrational?
The cube root of 343 is rational. A rational number can be written as a fraction of two integers, and since ∛343 = 7, which is a whole number, it is also a rational number.
6. What is the prime factorization of 343?
The prime factorization of 343 is 7 × 7 × 7 or 7³. This shows that 343 is made up of three identical prime factors, which confirms that it is a perfect cube and helps in finding its cube root easily.
7. What is the difference between square root and cube root of 343?
The square root of 343 is not a whole number, while the cube root of 343 is 7.
- Square root: A number that multiplies by itself twice (√343 ≈ 18.52, irrational).
- Cube root: A number that multiplies by itself three times (∛343 = 7).
8. How do you write the cube root of 343 in exponential form?
The cube root of 343 in exponential form is 3431/3. Using the law of exponents, since 343 = 7³, we get (7³)1/3 = 7. Therefore, 3431/3 equals 7.
9. Can you give an example to verify the cube root of 343?
Yes, you can verify it by multiplying 7 three times to get 343.
- 7 × 7 = 49
- 49 × 7 = 343
10. What are some common mistakes when finding the cube root of 343?
A common mistake is confusing cube root with square root, but the correct cube root of 343 is 7.
- Mixing up √343 with ∛343.
- Stopping factorization too early.
- Forgetting that cube root involves three identical factors.
