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Square Root of 1521
A perfect square number is the number obtained after squaring a whole number. It can also be defined as the number whose square root always results in a whole number. 1521 is such a number which is obtained after squaring 39. Thus, the square root of 1521 results in a whole number (39). This makes 1521 a perfect square number. Hence, the square root of 1521 is a rational number. In this minilesson, we will learn to find the square root of 1521 along with solved examples. Let us now find the square root of 1521.
 Square Root of 1521: √1521 = 39
 Square of 1521: 1521^{²} = 2,313,441
What Is the Square Root of 1521?
The square root of a number is a value which on squaring gives the original number. For example, the square root of 1521 (which is denoted by √1521) is 39.
 Square root is a number (integer) which when multiplied by itself results in the original number.
 1521 = a × a = 39^{²}
 Then a = √1521 = √(39 × 39)
 39 × 39 = 1521 or 39 × 39 = 1521
 The square roots of 1521 are +39 or 39.
 This shows that 1521 is a perfect square.
Is the Square Root of 1521 Rational or Irrational?
A rational number is defined as a number that can be expressed in the form of a quotient or division of two integers, i.e. p/q, where q is not equal to 0. A number that is not a rational number is called an irrational number. Nonterminating decimals, which have repeated numbers after the decimal point are rational numbers. Let us talk about the number 1521. The square root of 1521 can be expressed as 39/1 and 39/1. Both the numbers can be represented in the form of a rational number. Hence, the square root of 1521 is a rational number.
How to Find the Square Root of 1521?
Let us calculate the square root of 1521 using the prime factorization method.
Square Root of 1521 by Prime Factorization Method
 Prime factorization of 1521: 3 × 3 × 13 × 13
 Prime factors of 1521 in pairs: (3 × 3) × (13 × 13)
 Square root of 1521: (3)^{² }× (13)^{²} = ±39
 Therefore, √1521 = ±39
Important Notes:
 1521 is a number that is a perfect square. This indicates that it has a natural number as its square root.
 The square root of 1521 is 39.
 The square root of 1521 can be expressed as a fraction of the form p/q. This tells us that the square root of 1521 is a rational number.
 The square root of 1521 is either 39 or 39.
 39 and 39 can be expressed as 39/1 and 39/1.
 Both the numbers 39 and 39 can be represented in the form of a rational number.
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Challenging Questions
 What is the negative root of 15210?
 Find the square root of 152 up to 5 decimal places?
 What is the square root of:
a) √√√1521
b) √1521
Solved Examples

Example 1: Find the square root of 1521 using long division method.
Solution:
The square root of 1521 can be found using the long division method. Here are the steps:
 Step 1: We pair digits of the given number starting with the digits from the right. Place a horizontal bar over the numbers to indicate pairing.
 Step 2: Now we find a number which on multiplication with itself gives a product less than or equal to 15. As we know 3 × 3 = 9 < 15. Hence, the difference obtained is 6 and the quotient is 3.
 Step 3: Now, we have to bring down 21 and multiply the quotient by 2. This give us 6. Hence, 6 is the starting digit of the new divisor.
 Step 4: 9 is now placed at one's place of the new divisor. 69 multiplied by 9 gives 621. The obtained remainder is 0.
 Step 5: Hence, the square root of 1521 is 39.

Example 2: Rosie is multiplying a number by itself. If the product is 1521, help Rosie find the number. Find the square root of the number obtained.
Solution:
To find the number, let us assume the number = q
On multiplying q times q = q × q = 1521
q² = 1521
q = √1521
q = 39
(39 × 39 = 1521)
The number is 39.
The square root of 39 using long division method is:The square root of 39 is 6.244.

Example 3: Rosie wants to buy a new wall painting. In the store, she finds a square painting that has a side of 39 units. Calculate the area of the wall painting.
Solution:
Side of the square painting = 39 units
Area = side × side = 39 × 39 = 1521
Area of the wall painting is 1521 sq. units.
FAQs on Square Root of 1521
What is the square root of 1521 using prime factorization?
Prime factorization of 1521: 3 × 3 × 13 × 13
Prime factors of 1521 in pairs: (3 × 3) × (13 × 13)
Square root of 1521: (3)² × (13)² = ±39
Therefore, √1521 = ±39
Is 1521 a perfect square?
1521 is a perfect square. 1521 is a natural number. There is another natural number, which is 39, that can be squared to give 1521 as the result. Thus, 1521 is a perfect square.
What is the value of square root of 1521?
The value of the square root of 1521 is √1521 = 39.
Is the square root of 1521 a rational number?
Yes, the square root of 1521 is a rational number since the square root of 1521 can be represented in the form of p/q.
Can we find the square root of 1521 using the repeated subtraction method?
Yes, we can find the square root of 1521 by repeated subtraction method as it can be used only for perfect squares, and 1521 is a perfect square.
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