## Square & Square Root of 8

## Square of 8

**8² (8×8) = 64**

To calculate the square of 8, you simply multiply 8 by itself:

Therefore, the square of 8 is 64. This straightforward calculation is a building block for more complex mathematical operations and concepts, including algebraic equations, geometric formulas, and statistical models.

## Square Root of 8

**√8 = 2.8284271247461903**

**Or**

**√8 = 2.828 up to three places of decimal**

The square root of 8, symbolized as √8, holds a fascinating position in mathematics, showcasing the intriguing world of irrational numbers. When we talk about square roots, we refer to finding a number that, when squared (multiplied by itself), equals 8. The square root of 8 can also be represented in exponential form as 8^½ or 8^⁰.⁵ This number, precise to eight decimal places, is 2.82842712, and it serves as a positive solution to the equation x² = 8.

**Square Root of 8:** 2.8284271247461903

**Exponential Form:** 8^½ or 8^0.5

**Radical Form:** √8 or 2√2

## Is the Square Root of 8 Rational or Irrational?

The square root of 8 is an **irrational number**. This can be understood by breaking down what rational and irrational numbers are:

- A
**rational number**is any number that can be expressed as a fraction a/b where a and b are integers, and b is not equal to zero. It includes integers, fractions, and finite or repeating decimals. - An
**irrational number**is a number that cannot be expressed as a simple fraction. Irrational numbers have non-repeating, non-terminating decimal expansions.

To see why the square root of 8 is irrational, let’s first simplify it:

√8=√4⋅2=√4⋅√2=2√2

We know that 22 is a rational number. However, 22 is famously irrational. It cannot be represented as a fraction of two integers, and its decimal form goes on forever without repeating. Therefore, 2√2 is also irrational, because the product of a rational number (in this case, 22) and an irrational number (√2) is always irrational.

To summarize, the square root of 8 is irrational because it can be simplified to 2√2, which involves the square root of 2 – a well-known irrational number

## Method to Find Value of Root 8

To find the square root of 8, we can use several methods, including estimation, the prime factorization method, and using a calculator. Here, we focus on a simple estimation method for understanding:

**Identify Perfect Squares Around 8:**Recognize that the perfect square closest to but less than 8 is 4 (2²=4), and the perfect square closest to but more than 8 is 9 (3²=9).**Estimation**: Since 8 is between 4 and 9, its square root will be between 2 and 3.**Refinement**: To get a more precise value, use a calculator or a square root table. The square root of 8 is approximately 2.8284271247461903.**Simplified Radical Form:**The square root of 8 can also be expressed in simplified radical form as √8=√4×2=2√2.

## Square Root of 8 by Long Division Method

**Step 1: Preparation**

Write 8 as 8.00 00 00, grouping digits in pairs from the decimal point. For 8, it looks like “08”.

**Step 2: Find the Largest Square**

Identify the largest square smaller than or equal to 8, which is 4 (2²). Place 2 above the line as the first digit of the root.

**Step 3: Subtract and Bring Down**

Subtract 4 from 8 to get 4, then bring down the next pair of zeros to make it 400.

**Step 4: Double and Find the Next Digit**

Double the current result (2) to get 4. Now, find a digit (X) such that 4X multiplied by X is less than or equal to 400. Here, X is 8, because 48×8=384

**Step 5: Repeat with Precision**

Subtract 384 from 400 to get 16, bring down the next zeros to get 1600, then double the quotient (28) to get 56. Choose a digit (Y) so 56Y×Y is just under 1600.

**Step 6: Finish at Desired Accuracy**

Continue the process until reaching the desired level of accuracy. For the square root of 8, this method gives us about 2.828 as we extend the division.