## Square & Square Root of 343

## Square of 343

**343² (343 × 343) =117649**

The square of 343, a square number, is 117,649. Squaring 343 entails multiplying it by itself, showcasing a fundamental mathematical operation. Understanding the properties of square numbers like 343 aids in comprehending mathematical concepts and relationships, contributing to a deeper understanding of numerical systems and their applications.

## Square Root of 343

**√343 = 18.520259**

The square root of 343, an irrational number, is approximately 18.520259177452136. It represents the number that, when multiplied by itself, equals 343. Understanding square roots illuminates mathematical principles, offering insights into numerical relationships and facilitating problem-solving across various disciplines.

## Is the Square Root of 343 Rational or Irrational?

**The square root of 343 is an irrational number**

The square root of 343 is irrational. This is because 343 is not a perfect square, meaning there is no integers that, when multiplied by itself, equals 343. The square root of 343 is approximately 18.520259177452136, which cannot be expressed as a simple fraction and has a non-repeating, non-terminating decimal expansion. Thus, it is classified as an irrational number.

Rational numbers can be expressed as a fraction of two integers, where the denominator is not zero.

Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions.

## Methods to Find Value of Root 343

1. **Prime Factorization Method**:

- Factorize 343 into prime factors: ( 343 = 7 × 7 × 7 ).
- Since the factors are not paired completely, it indicates that 343 is not a perfect square.
- Approximate the square root using other methods or a calculator.

2. **Estimation Method**:

- Identify perfect squares close to 343, such as 324 (18²) and 361 (18²).
- Estimate the square root to be between 18 and 19.
- Refine the estimate by averaging or iterative approximation.

3. **Using a Calculator**:

- Use a scientific calculator to find the square root directly.
- Enter 343 and press the square root (√) button to get approximately 18.520259.

4. **Newton’s Method (Iterative Method)**:

- Start with an initial guess (e.g., 18).
- Use the formula ( xₙ₊₁} = 1/2( xₙ + 343/xₙ).
- Repeat the iteration until the value stabilizes around 18.520259.

Each method offers a way to understand and approximate the square root of 343 with varying degrees of precision.

## Square Root of 343 by Long Division Method

**Step 1:**

Write 343.000000. Take the number in pairs from the right. 1 stands alone. Now, divide 3 by a number such that (number × number) gives ≤ 3. Obtain quotient = 1 and remainder = 1. Double the quotient. We get 2. Have 20 as our new divisor. Bring down 43 for division.

**Step 2:**

Find a number such that (20 + that number) × that number gives the product ≤ 243. We find that 28 × 8 = 224. Subtract from 243 and get 19 as the remainder. Bring down the pair of zeros. 1900 is our new divisor.

**Step 3:**

18 is our quotient, and on doubling it becomes 36, and 360 is our new divisor. Find a number such that (360 + the number) × number gets 1900 or less than that. We find 365 × 5 = 1825. Subtract this from 1900 and get the remainder, and bring down the zeros. 7500 becomes the new dividend.

Repeat the steps until we approximate the square root to 3 decimal places. √343 = 18.520

## 343 is Perfect Square root or Not

A perfect square is a number that can be expressed as the square of an integer. When you take the square root of 343, you get approximately 18.52, which is not an integer. Therefore, 343 is not a perfect square.

## FAQs

## What is the root of 343 in simplest form?

The root of 343 in simplest form is 7, as 7 multiplied by itself equals 343.

## What is 3 to square root of 343?

3√343 is equal to 3^7, which is 21872187

## Is 343 a Perfect Cube?

Yes, 343 is a perfect cube because it can be expressed as 7373.

## What are the factors of 343?

The factors of 343 are 1, 7, 49, and 343, since these numbers divide 343 evenly without leaving a remainder.

## Is 343 a cube root?

343 is not a cube root. A cube root of a number is a value that, when multiplied by itself three times, equals the original number.