Root 5 – Calculation of Square Root Value

Root 5 Value

The root of a numeric value ‘x’ is equal to another numeric value, when multiplied by itself ‘n’ number of times, equals to the numeric value ‘x’ for which we are finding the root. So, we can also say that the square root of a number is just the inverse of squaring it. In this article, we will learn how to calculate the value of root 5.

Let us look at the root in correlation with power.

Consider the number’ a’ and a natural number ‘x.’ 

Then, a^x, = a x a x a …… (multiplied ‘x’ number of times)

If the value of ‘x’ is 2 and ‘a’ is 4, then 4² = 4 x 4 = 16

This means, ‘4 raised to the power 2′ is 16.

Now, the root is the inverse of power. Therefore, we can also see that 4, when multiplied by itself, yields 16. Hence, if 16 is the square of 4 then 4 is also the square root of 16.

We can confirm that amidst the long list of irrational numbers, it also includes square root 5. The root of 5 cannot be represented as a fraction, and it has an endless number of decimals places. Therefore, deeming it to be an irrational number.

The value of the square root of 5, reduced to 10 decimal places is 2.2360679774…

How can you find the value of root 5?

To determine a number’s square root, we generally verify whether it is a perfect square or not. The value of perfect squares is easy to locate, but we have to apply the long division method to get a root value for non-perfect squares. 

For example, a number like 2, 3, 5, 20, etc. is not a perfect square; however, 4, 9, 25 etc., are perfect squares, giving a whole number when calculating their root.

How to find the square root by division method?

Step 1: Put a bar on every pair of numerals starting from right to left. Since we have an odd number of digits, the extreme left single digit, i.e., 5, has a bar as well.

Step 2: Think of a number whose square is less than or equal to 5. In this case, we have 2 as its square is equal to 4.

Step 3: Take the number 2, as the new divisor and quotient. Now, divide to the next remaining number below the extreme left.

Step 4: Bring down the next pair of numbers to the right of the remainder under the next bar.

Step 5: To calculate the divisor, multiply the previous quotient by 2 and choose a number such that the number is less than or equal to the new dividend.

Step 6: Keep repeating steps 2, 3, 4 and 5 until the remainder is 0 or repetitive.

Step 7: The quotient you thus obtain will be the value of the square root by long division method. 

Thus, the root 5 value up to 2 decimal places us 2.24

Represent root 5 on number line

To represent √5 on the number line, consider drawing a number line as drawn in the image. Now, mark a point at 2 units as A. Draw a perpendicular AX at A. then cut off an arc AB = 1 unit

Now, we have OA = 2 units and AB = 1 unit

Using Pythagoras theorem,

OB2 = AB² + OA²

OB2 = 2² + 1²

OB2 = 5

OB = √5

Now, since OB is equal to root 5, by taking O as the centre, draw an arc that cuts the number line at C. Clearly, OC = OB = √5

Hence, point C represents root 5 on number line.

Examples

Example 1: Prove that root 5 is irrational.

To prove the above statement, we must assume that √5 is a rational number, i.e., we can express it as p/q
⇒ √5 = p/q

here p and q are co-prime integers, and q≠0

Squaring both the sides, we get,
⇒5 = p²/q²
⇒5q² = p² —————–(i)

p²/5 = q²

So, if 5 divides p, then p is a multiple of 5.

⇒ p = 5m
⇒ p² = 25m² ————-(ii)
From equations (i) and (ii), we get,
5q² = 25m²
⇒ q² = 5m²
⇒ q² is a multiple of 5
therefore, q is also a multiple of 5
Hence, the common factor of p and q is 5. 

So, p/q is not a rational number

Hence proved, √5 is an irrational number.

Example 2: Calculate the value of 1/√5

By rationalizing the denominator, we can find the value of 15

So,

= (1/√5)* (√5/√5)

= (1* √5) / (5* √5)

= √5 / (√5)²

We know that the square of root 5, i.e., root 5 into root 5 is equal to 5

= √5/5

Therefore, the value of 1/root 5 is equal to √5/5