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Square Root of 73

The square root of 73 is expressed as √73 in the radical form and as (73)½ or (73)0.5 in the exponent form. The square root of 73 rounded up to 7 decimal places is 8.5440037. It is the positive solution of the equation x2 = 73.

  • Square Root of 73: 8.54400374531753
  • Square Root of 73 in exponential form: (73)½ or (73)0.5
  • Square Root of 73 in radical form: √73
1. What Is the Square Root of 73?
2. Is Square Root of 73 Rational or Irrational?
3. How to Find the Square Root of 73?
4. Important Notes
5. Thinking Out of the Box!
6. FAQs on Square Root of 73

What Is the Square Root of 73?

The square root of 73 is the number whose square gives the original number (73).  There are no integers whose square will give the value 73.

√73 = 8.544

On squaring 8.544, we get (8.544)2 = 72.999936... which is very close to 73 when it is rounded to its nearest value.

Is the Square Root of 73 Rational or Irrational?

For a number to be a  rational number, it should either be terminating or non-terminating and have a repeating pattern in its decimal part. We saw that √73 = 8.544003745317531...This is non-terminating and the decimal part has no repeating pattern. Hence, √73 is an irrational number.

How to Find the Square Root of 73?

There are different methods to find the square root of any number. Click here to know more about it.

Simplified Radical Form of Square Root of 73

As 73 is a prime number, it has only two factors, 1 and 73. Thus, the simplified radical form of √73 is √73.

Square Root of 73 by Long Division Method

The square root of 73 can be found using the long division as follows.

  • Step 1: We pair digits of a given number starting with a digit at one's place. Put a horizontal bar to indicate pairing.
  • Step 2: Now we find a number which on multiplication with itself gives a product less than or equal to 73. As we know 8 × 8 = 64 < 73. Hence, their difference gives 9 and the quotient is 8. 
  • Step 3: Now, we have to bring down 00 and multiply the quotient by 2. This give us 16. Hence, 16 is the starting digit of the new divisor.
  • Step 4: 5 is placed at one's place of new divisor because when 165 is multiplied by 5 we get 825. The obtained answer now is 75 and we bring down 00.
  • Step 5: The quotient now becomes 85 and it is multiplied by 2. This gives 170, which then would become the starting digit of the new divisor.
  • Step 6: 4 is placed at one's place of new divisor because on multiplying 1704 by 4 we get 6816. The new divisor now becomes 684 and we bring 00 down.
  • Step 7: Now the quotient is 854. When multiplied by 2, it gives 1708, which will be the starting digit of the new divisor.
  • Step 8: 4 is placed at one's place of the divisor because on multiplying 17084 by 4 we will get 68336. So, now the divisor is 64.

Square Root of 73 by Long Division

So far we have got √73 = 8.544. On repeating this process further, we get, √73 = 8.544003745317531...

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Important Notes:

  • 73 lies between 64 and 81. So √73 lies between √64 and √81, i.e., √73 lies between 8 and 9.
  • The square root of 73 is irrational as 73 is not a perfect square, which makes it difficult to simplify √73 further.

Think Tank:

  • Is the value of √(-73) same as -√73?
  • Is √-73 a real number? 
 

Square Root of 73 Solved Examples

  1. Example 1: What is the length of a square room having an area 73 square feet?

    Solution

    Let us assume that the length of the room is x feet. Then the area of the room's floor is x2 square feet. By the given information:

    x2 = 73
    x = √73 = 8.5 feet (approx.)

    Hence, the length of the room is 8.5 feet.

  2. Example 2: Will the length of radius of a circle be less than 10 inches or more than 10 inches if the area of circle is 73π square inches?

    Solution

    Area is found using the formula: area of a circle = πr2 square inches. By the given information,

    πr2 = 73π 
    r2 = 73

    By taking the square root on both sides, √r2= √73. We know that the square root of r2 is r.
    By calculating the square root of 73 is 8.54 inches. Hence, the length of radius of circle is less than 10 inches if the area of circle is 73π square inches.

  3. Example: Solve the equation x2 − 73 = 0

    Solution:

    x2 - 73 = 0 i.e. x2 = 73
    x = ±√73
    Since the value of the square root of 73 is 8.544,
    ⇒ x = +√73 or -√73 = 8.544 or -8.544.

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FAQs on the Square Root of 73

What is the Value of the Square Root of 73?

The square root of 73 is 8.544.

Why is the Square Root of 73 an Irrational Number?

The number 73 is prime. This implies that the number 73 is pairless and is not in the power of 2. Therefore, the square root of 73 is irrational.

Is the number 73 a Perfect Square?

The number 73 is prime. This implies that the square root of 73 cannot be expressed as a product of two equal integers. Therefore, the number 73 is not a perfect square.

What is the Square of the Square Root of 73?

The square of the square root of 73 is the number 73 itself i.e. (√73)2 = (73)2/2 = 73.

What is the Square Root of -73?

The square root of -73 is an imaginary number. It can be written as √-73 = √-1 × √73 = i √73 = 8.544i
where i = √-1 and it is called the imaginary unit.

Evaluate 9 plus 2 square root 73

The given expression is 9 + 2 √73. We know that the square root of 73 is 8.544. Therefore, 9 + 2 √73 = 9 + 2 × 8.544 = 9 + 17.088 = 26.088

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