What is the smallest number by which 288 should be divided to get a perfect square?

The square root of a number is the value when multiplied by itself gives the original number. e.g. 3 × 3 = 9 so, the square root of 9 is 3. Once you grasp, the basic concept of square roots, then any lesson will be easily solved. As per mathematical historian D.E. Smith, Aryabhata's method for finding out the square root was introduced in Europe by Cataneo in 1546. The Babylonians were the first to find the simplest method for finding square roots of numbers. It was also called Heron's method. In this mini-lesson, we will learn about the square root of 288. We will do so using methods like prime factorization and long division method.

• Square Root of 288: √288 = 16.970
• Square of 288: 2882 = 82,944

What Is the Square Root of 288?

The square root of a number is the number that gets multiplied to itself to give the product. For any two real numbers a and b,

a² = b
a = √b

The above expression means that a is the 2nd root or square root of b. The square root of 288 means that number which when multiplied with itself will give the result as 288. The definition above can be represented as,
Square root of 288 = √288.
The square root of 288 is the value obtained after performing the operation of the square root on 288.

Is the Square Root of 288 Rational or Irrational?

288 cannot be broken into two factors which on multiplying give 288. It can be approximately written as a square of 16.97, which is a non-recurring and non-terminating decimal number. This shows it isn't a perfect square, which also proves that the square root of 288 is an irrational number.

How to Find the Square Root of 288?

The square root of 288 is found using the following steps:

• Check whether the number is a perfect square or not. 288 is not a perfect square as it cannot be broken down into a product of two same numbers.
• If the number is a perfect square, it can be written in this format √x2 = x
• If the number is not a perfect square, the square root is found using the long division method. It can also be written in its simplified radical form of a square root.
• In this case, 288 is not a perfect square. Hence, its square root is found using the long division method. The simplified radical form of the square root of 288 is given below.

Simplified Radical Form of Square Root of 288

288 can be written as a product 2 and 144. It is given as √288 = √2 × 144. 144 is a perfect square of 12. Hence, the square root of 144 is 12. 2 is not a perfect square, hence it remains within roots.
√288 = 12 × √2
Simplified radical form of the square root of 288 is 12√2

Square Root of 288 By Long Division

Let us understand the process of finding the square root of 288 by long division.

• Step 1: Pair the digits of the number starting from one's digit. 288 has 3 digits. When 2 digits are paired from the right side, 88 forms a pair while 2 aren't paired. We show the pair by placing a bar over them.
• Step 2: Now we need to find a number such that the square of the number gives a result less than or equal to the first pair from the left side. Here, the first pair from the left side just consists of 1 number, i.e., 2. A Square of 1 gives a product less than 2. The number is subtracted from the first pair and subsequently, the next pair is added as the divisor, i.e., 88
• Step 3: Now we take the double of quotient and place a digit with divisor along with its placement in the quotient, such that the new divisor, when multiplied with the individual number in the quotient, gives the product less than the dividend subsequently subtracting it from the dividend. The double of 1 is 2. Now a number is to be placed along with 2 such that the product of the two-digit number with the quotient gives a product less than 188. The product of 26 by 6 gives 156. When the difference between 156 and 188 is calculated, the obtained value is 32
• Step 4: For the new dividend obtained, we take the double of quotient and place a digit with divisor along with its placement in the quotient, such that the new divisor, when multiplied with the individual number in the quotient, gives the product as less than the dividend. Now double of the quotient is taken, i.e., 32. Now a decimal is added to the quotient, hence letting us add a pair of zeros to the original dividend. A number along with 32 is added in the blank and it is multiplied by the same number in the quotient. Product of 329 with 9 gives 2961
• Step 5: The difference is obtained in the above step. The double of the quotient is again taken and used as a divisor along with the involvement of one more digit such that the same digit is mentioned in the quotient, resulting in a product less than the new divisor. The difference obtained from the above step is 239. The double of the quotient, i.e., 169 is taken, which makes the divisor 338 along with the involvement of 7. The product of 3387 with 7 gives 23709. The difference between 23900 and 23709 is 191
• Step 6: The process is repeated. Hence, the division is shown as:

Explore Square roots using illustrations and interactive examples

Important Notes:

• If 288 was a perfect square, the square root of 288 would be a rational number. This concludes that the square root of any number "n," which is not a perfect square, will always be an irrational number.

Think Tank:

• How will Maria find the square root of 128 using the long division method?
• Can you find the square root of 288 using the long division method up to 3 decimal places?

1. Example 1: Help Angelica find the two consecutive numbers between which the square root of 288 lies.

   Solution

   Angelica knows that the perfect squares nearest to 288 are 289 and 256 The square root of 289 is 17 The square root of 256 is 16 These are the two numbers between which the square root of 288 lies.

   Hence, Angelica found that the square root of 288 lies between 16 and 17

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FAQs On Square Root of 288

What is the square root of 288?

The square root of 288 is 16.970.

Why is √288 an irrational number?

A number with decimal expansion as non-terminating and non-repeating is always an irrational number. So, √288 is an irrational number.

How do you find the square root of 288?

We can find the square root of 288 by using the prime factorization and long division.

Is the square root of 288 infinite?

The decimal expansion of √288 is infinite because it is non-terminating and non-repeating.
The square root of 16 is 4, which is a rational number.

What is the square root of 288 in simplified form?

The square root of 288 in simplified form is 12√2