Square root of 33

The square root is a number that, when multiplied by itself, results in the original number whose square root is to be found. Know that the square root of 33 is ±5.74456264654.

We will see here more about the square root of 33. As defined, the square root is just the opposite (inverse) of squaring a number, so, squaring 5.744… will result in 33. The positive value, 5.744… is the solution of the equation x² = 33.

It contains both positive and a negative root, where the positive root is called the principal square root. The square root of 33 is expressed as √33 in radical form. In exponential form, it is written as (33)^1/2  .

We now came to a point where we can say that:

• 33 is a non-perfect square.

• √33 is an irrational number.

Let us now find how we got this value of 5.744… as a square root of 33.

We will use these methods below to find.

•  Prime factorization method

•  Long division method

• Approximation/Estimation method

The prime factorization of 33 involves breaking down a number into its factors.

Factorize 33 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. 

• Find the prime factors of 33.

• After factoring 33, make pairs out of the factors to get the square root.

• If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

Prime factorization of 33 = 33×1

For 33, no pairs of factors are obtained, so a single 33 is remaining.

So, it can be expressed as √33 = √(33×1) = √33.

√33 is the simplest radical form of √33.

Long Division method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder for non-perfect squares. To make it simple it is operated on divide, multiply, subtract, bring down and do-again.

To calculate the square root of 33:

Step 1: On the number 33.000000, draw a horizontal bar above the pair of digits from right to left.

Step 2 :Find the greatest number whose square is less than or equal to 33. Here, it is 5, Because 5²=25 < 33.

Step 3 : Now divide 33 by 5 such that we get 5 as a quotient and then multiply the divisor with the quotient, we get 25. Add a decimal point after the new quotient, 5.

Step 4: Subtract 25 from 33. Bring down two zeros and place it beside the difference 8.

Step 5: Add 5 to the same divisor, 5. We get 10.

Step 6: Now choose a number such that when placed at the end of 10, a 3-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 800. Here, that number is 7. 

107×7=749<800. In quotient’s place, we also place that 7.

Step 7: Subtract 800-749=51. Again, bring down two zeroes and make 51 as 5100. Simultaneously add the unit’s place digit of 107, i.e., 7 with 107. We got here, 114. Apply Step 5 again and again until you reach 0. 

We will show two places of precision here, and so, we are left with the remainder, 6464 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 8 : The quotient obtained is the square root. In this case, it is 5.744….

Estimation of square root is not the exact square root, but it is an estimate, or you can consider it as a guess.

Follow the steps below:

Step 1: Find the nearest perfect square number to 33. Here, it is 25 and 36.

Step 2: We know that, √25=±5 and √36=±6. This implies that √33 lies between 5 and 6.

Step 3: Now we need to check √33 is closer to 5.5 or 6. Since (5.5)²=30.25 and (6)²=36. Thus, √33 lies between 5.5 and 6.

Step 4: Again considering precisely, we see that  √33 lies close to (5.5)²=30.25. Find squares of (5.6)²=31.36 and (5.8)²= 33.64.

We can iterate the process and check between the squares of 5.7 and 5.78 and so on.

We observe that √33 = 5.744…
 

• Negative Square root - The negative square root of 33 is √-33.

• GPS - GPS or Global Positioning System is basically a network of satellites used to determine the location of a point on Earth.

• Exponential form - An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors, or a way to write a number that is multiplied by itself more than one time.  Ex: 5× 5 × 5 × 5 = 625 Or, 5⁴ = 625, where 5 is the base, 4 is the exponent 

• Prime Factorization - Expressing the given expression as a product of its factors Ex: 52=2 × 2 × 13 

• Prime Numbers - Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers - The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers

• Irrational numbers - Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect square numbers - Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25.

• Non-Perfect square numbers -  Those numbers whose square roots comprise decimal places. Ex : 3, 10, 29