Square Root of 46
The square root of 46, also known just as root 46, is the number which when multiplied by itself results in the number 46. To find the length of a side of a square with area equal to 46, we need to know its square root. We will now see how to calculate the square root of 46 and discover few other interesting facts.
Square Root of 46: √46= 6.78232998...
Square of 46: 46² = 2116
1.  What Is the Square Root of 46? 
2.  Is Square Root of 46 Rational or Irrational? 
3.  How to Find the Square Root of 46? 
4.  Important Notes 
5.  Challenging Questions 
6.  FAQs on Square Root of 46 
What Is the Square Root of 46?
The square root of a number is a number which when multiplied by itself, results in the original number. For example, the square root of 25 is 5, as 5 times 5 results in 25 which makes it a perfect square. However, you can also have square roots of some numbers that do not result in whole numbers, such as 46. We can express the square root of 46 in different ways.
 Decimal form: 6.782
 Radical form: √46
 Exponent form: 46^{1/2}
Is the Square Root of 46 Rational or Irrational?
 The decimal part of the square root 46 is nonterminating. This is the definition of an irrational number.
 Looking at the decimal form of the root 46, we see that it is neverending.
√46= 6.78232998…….  Therefore, we can conclude that square root of 46 is irrational.
How to Find the Square Root of 46?
Since we concluded that the square root of 46 is nonterminating, we can only use 2 methods to calculate the value of its square root.
 Long Division Method
 Estimation and Approximation Method
Long Division
 Step 1: Starting from the right, we will pair up the digits of 46 by putting a bar above them. We also pair the 0s in decimals from left to right.
 Step 2: Think of a number whose square is less than or equal to 46. In this case, that number would be 6.
 Step 3: Dividing 46 by 6 with quotient set as 6, we get a remainder of 10.
 Step 4: We bring down a pair of 0s to the remainder and therefore get 1000.
 Step 5: The divisor, in this case 6 is doubled and written below. Now, we have 12X as the new divisor, and we need to find a value of X that makes the product of 12X × X less than or equal to 1000. In this case, 127 is the required value.
 Step 6: The number 7 is placed in the quotient after a decimal place. The new divisor for the next division will be 12X + X, in this case, 134.
Proceeding in the same manner and repeating from step 4, we can calculate the rest of the decimals.
Therefore, the square root of 46 = 6.782
Estimation and Approximation
The estimation method gives us an approximate answer and is usually not accurate to more than 1 decimal place. However, it is easy to perform as can be seen under.
 Step 1: Find a perfect square that is smaller than and bigger than 46. In this case, 6 and 7 will work as their squares are 36 and 49.
 Step 2: Writing in terms of inequality 6<√46<7 = 36<26<49
 Step 3: Multiply by 10 and write in terms of square roots: √3600<√4600<√4900
 Step 4: Move closer to inequality: √4489<√4600<√4624 = 67<10√46<68
= 6.7<√26<6.8  Step 5: Taking average of upper and lower limits we get (6.7 + 6.8)/2 = 6.75
Therefore, we can estimate the square root of 46 ≅ 6.75
Explore square roots using illustrations and interactive examples
Important Notes
 The real roots of √46 are ± 6.782
 Square root of a perfect square is always a whole number and that of other numbers is always irrational. For example, √16 = 4, whereas √17 = 4.1231…
Challenging Questions
 Find the value of √√46.
 What are the roots of 46?
 Determine the square root of 4646.
Solved Examples

Example 1: Dave was wondering whether the value of √46 is the same as √46. What do you think?
Solution:
Square roots of negative numbers cannot be real numbers.
√46 is a real number. However, √46 is an imaginary number.
Hence, they are not the same, and √46 is not the same as √46. 
Example 2: What is the perimeter of a square whose area is 46 cm²?
Solution:
Let area be denoted by A, side length by s, and perimeter by P
P = 4 × Length of side
A = (Length of side)² = s²
Therefore s = √A = √46
P = 4s = 4 × √46 = 27.129 cmTherefore, the perimeter of the land is about 27.13 cm.
FAQs on Square Root of 46
What is the square root of 46?
The square root of 46 is √46= 6.782.
Can the root of 46 be negative?
The root of the number 46 denotes the positive and negative value of 6.782.
Therefore, the root of 46 can be negative.
What is the square root of 46 in the simplest radical form?
√46 is the simplest radical form.
What is the square of 46?
The square of 46 is 46^{2} = 2116
Can we find the square root of 46 by the prime factorization method?
No, we can’t find the square root of 46 by prime factorization method. This is because its factors are all prime numbers (2 and 23) with power 1 and therefore we cannot simplify it further.
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