Square Root of 97

Did you know 97 is not a perfect square, where the sum of its digits is a perfect square? In this mini-lesson we will learn to find square root of 97 by long division method along with solved examples. Let us see what the square root of 97 is.

• Square Root of 97: √97 = 9.848
• Square of 97: 97² = 9409

The square root of 97 is a number whose square gives the original number. What number that could be? By trial and error method, we can see that there does not exist any integer whose square is 97.

√97 = 9.848

To check this answer, find (9.848)² and we can see that we get a number 96.983104... which is very close to 97.

A rational number is a number which is:

• either terminating
• or non-terminating and has a repeating pattern in its decimal part.

In the previous section, we saw that: √97 = 9.8488578018... Clearly, this is non-terminating and the decimal part has no repeating pattern. So it is NOT a rational number. Thus, √97 is an irrational number.

We can find the square root of 97 using various methods.

1. Repeated Subtraction 
2. Prime Factorization
3. Estimation and Approximation
4. Long Division

If you want to learn more about each of these methods, click here.

Simplified Radical Form of Square Root of 97

97 is a prime number and thus it has only two factors, 1 and 97 itself. To find the square root of any number, we take one number from each pair of the same numbers from its prime factorization and we multiply them. But the factorization of 97 is 1 × 97 which has no pairs of the same numbers. Thus, the simplest radical form of √97 is √97 itself.

Square Root of 97 by Long Division Method

The square root of 97 can be found using the long division as follows.

• Step 1: Pair of digits of a given number starting with a digit at one's place. Put a horizontal bar to indicate pairing.
• Step 2: Now we will find a number which when multiplied to itself gives a product of less than or equal to 97. We know 9 × 9 = 81 ≤ 97. Thus the divisor is 9 and the quotient is 9. Now proceed with the division.
• Step 3: Now, we have to bring down 00 and multiply the quotient by 2 which would give us 18. 18 is the starting digit of the new divisor.
• Step 4: 8 would be placed at one's place of new divisor because when 188 is multiplied by 8 we will get 1504. So, the new divisor is 188 and the next digit of the quotient is 8. Now we would proceed with the division and get the remainder.
• Step 5: Next, we have to bring down 00, and quotient 98 is multiplied by 2 will give 196, which then would become the starting digit of the new divisor.
• Step 6: 4 will be placed at one's place of new divisor because on multiplying1964 by 4 we will get 7856. So, the new divisor is 1964 and the next digit of the quotient is 4. Now proceeding with the division to get the remainder.
• Step 7: Next, we will bring down 00 and quotient 984 when multiplied by 2 gives 1968, which will be the starting digit of the new divisor.
• Step 8: 8 will be placed at one's place of new divisor because on multiplying 19688 by 8 we will get 157504. So, the new divisor is 19688 and the next digit of the quotient is 8. Now proceeding with the division to get the remainder.

So far we have got √97 = 9.848. If we repeat this process further, we get, √97 = 9.8488578018...

Explore square roots using illustrations and interactive examples

• Square Root of 98
• Square Root of 89
• Square Root of 85
• Square Root of 200
• Square Root of 192

Important Notes:

• 97 lies between 81 and 100. So √97 lies between √81 and √100, i.e., √97 lies between 9 and 10.
• The prime factorization method is used to write a square root of a non-perfect square number in the simplest radical form. For example: 45 = 3 × 3 × 5 = 3² × 5. So, √45 = √3² × √5 = 3√5.

Think Tank:

• Can the value of a square root be negative as well? Hint: Think what is the square of a negative number.
• Is √-97 a real number? Hint: Think whether there is any real number whose square is negative.

Can the square root of 97 be simplified?

No, the square root of 97 cannot be simplified.

What is the square root of 97 rounded to its nearest tenth?

To round the square root of 97 to its nearest tenth means to have one digit after the decimal point to get the answer.
√97 = 9.8488 can be rounded to its nearest tenth as 9.8.

Does 97 have a square root?

97 can be just written as √97 as it cannot be simplified further as it is not a perfect square.

What is the square root of 97 simplified?

97 is not a perfect square and hence its square root is not a whole number. √97 = 9.848857 (approx.)

Is the square root of 97 rational or irrational?

The square root of 97 is irrational.

Is square root of 97 a real number?

Yes, the square root of 97 is a real number.