## Square & Square Root of 313

In the realm of mathematics, especially within algebra, squares and square roots are fundamental. They underpin our comprehension of rational and irrational numbers. Considering 313 squared, we delve into mathematical relationships and patterns.

## Square of 313

**313² (313× 313) = 97969**

The square number of 313 is 97969. This showcases the fundamental algebraic operation of squaring a number.

## Square Root of 313

**√313 ≈ 17.6918060**

**or**

**√313 ≈ 17.691 Upto 3 decimals**

Mastering square roots involves identifying the number that, when multiplied by itself, equals 313. This understanding is crucial for exploring algebraic relationships and patterns.

**Square Root of 313:**17.6918060

**Exponential Form:** 313^1/2 or 313^0.5

**Radical Form:** **√**313

## Is the Square Root of 313 Rational or Irrational?

**The square root of 313 is irrational.**

As it’s not a perfect square, it’s irrational. It can’t be expressed as a fraction of two integers.

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.

## Methods to Find the Value of Root 313

**Estimation Method**: Use initial approximation, refine iteratively.**Prime Factorization Method**: Express 313 as product of prime factors, then find square root.**Calculator**: Use square root function for direct computation.

## Square Root of 313 by Long Division Method

**Step 1:** Starting with 313.000000, divide 3 by a number such that (number × number) is ≤ 3. Obtain quotient = 1 and remainder = 1. Double the quotient, giving us 2. Our new divisor is 27. Bring down 13 for division.

**Step 2:** Find a number such that (2 + that number) × that number gives a product ≤ 213. We find that 27 × 7 = 189. Subtract from 213, resulting in a remainder of 24. Bring down the pair of zeros, making 2400 the new divisor.

**Step 3:** With 17 as our quotient, doubling it becomes 34, and 2400 is our new divisor. Find a number such that the number, when multiplied by itself and added to 34, gives a product ≤ 2400. We find 346 × 6 = 2076. Subtracting this from 2400, we get 324 as the remainder. Bring down the zeros, making 32400 the new dividend. Repeat until we approximate the square root to 3 decimal places. We get, √313 = 17.691.

## 313 is a Perfect Square or Not

**No, 313 is not a perfect square number.**

A perfect square cannot be expressed as the product of an integer multiplied by itself. It does not have an integer square root.

## FAQs

## How does the square root of 313 relate to Pythagorean triples?

The square root of 313 is not part of any Pythagorean triple, as it does not form the hypotenuse of a right triangle with integer sides.

## What are the factors of 313?

The factors of 313 are 1 and 313 because it is a prime number.

## Can the square root of 313 be simplified further?

No, the square root of 313 cannot be simplified further as it is already in its simplest form.