Square root of 18

Square root of 18 is equal to 3√2 in radical form and 4.24264068712 in decimal form. When we multiply the value of the square root of a number to itself, it produces the original number. Therefore, square root is the inverse or reverse process of squaring a number. If √18 = x, then x² = 18.

The square root of 18 is denoted by √18, where ‘√’ is the radical symbol and 18 is the radicand.

Since, √2 is an irrational number, therefore, √18 = 3√2 is also an irrational number. Therefore, we cannot represent the square root of the irrational number in the form of P/Q, where P is the numerator and Q is the denominator.

The number 18 is not a perfect square, hence the square root of 18 will not be a whole number. Also, we cannot determine the exact value of √18, using prime factorisation. Therefore, we can use the long division method to find the value of √18.

How to Find the Square root of 18?

We can find the square root using these methods:

• Prime Factorisation Method
• Long Division Method

Prime Factorisation Method

By the prime factorisation method, we can write:

18 = 2 x 3 x 3

Now, let us see how many prime factors can be paired to make square terms.

18 = 2 x (3 x 3)

Now, we can represent 3 x 3 = 3². Hence,

18 = 2 x 3²

Taking square root on both the sides, we get;

We can cancel the square term with the square root.

√18 = 3√2

Thus, this is the value of √18, in radical form.

Since, the value of √2 = 1.4142135624

Therefore, 3√2 = 3 x 1.4142135624 = 4.24264068712

Finally, the value of square root of 18 is:

√18 = 4.24264068712

Which is an irrational number.

Long Division Method

This is the quick method to find the square root of 18, accurately. We can use this method to determine the value of √18 up to four places of decimals. Let us see the steps that are involved in the long division method.

• Step 1: We can write the number 18 as 18.00 00 00 00
• Step 2: Now, we have to group the numbers in a pair of two.
• Step 3: We need to divide 18 by such a number that when it is multiplied by itself, it generates a value less than or equal to 18. Thus, 4 x 4 = 16
• Step 4: After dividing 18 by 4, we are left with a remainder equal to 2 and the divisor will become 8. Now, take down the next pair of numbers and combine with 2 to get 200, as shown in the figure.
• Step 5: Again combining the divisor 8 with such a number, such that the multiplication will result in a value less than or equal to the new dividend, i.e. 200.

82 x 2 = 164

• Step 6: Now, the divisor is 82 + 2 = 84 and remainder is 36. Repeat the method again up to four places of decimal.

Therefore, the square root of 18 is equal to 4.2426, approximately.

Approximation Method

Another most unlikely method of finding the square root of any number is the approximation method. With this method, we can find the approximate square root of any number but not the exact value.

Suppose we are finding the square root of 18 using this method, then:

18 lies between 4² = 16 and 5² = 25

Thus, the square root of 18 is more than 4 but less than 5, so we got the initial value.

Now, we need to find the decimal part.

Decimal part = (Actual number- Lower perfect square) /(Higher perfect square-Lower perfect square)

= (18-16)/(25-16)

= 2/9

= 0.22

Therefore, the value of √18 = 4.22

Repeated Subtraction Method

We cannot use the repeated subtraction method here to find the square root of 18. This method is applicable only if a number is a perfect square but 18 is an imperfect square. To check we can try this method here.

Basically, in this method, the actual number is repeatedly subtracted with the consecutive odd numbers, till we get a difference equal to zero. Starting with the actual number, we subtract it with 1, then the resulting value is subtracted with the next odd number, i.e. 3 and so on.

• 18 – 1 = 17
• 17 – 3 = 14
• 14 – 5 = 9
• 9 – 7 = 2
• 2 – 9 = -7

Since, we didn’t get zero in any step of the subtractions done above, therefore, we cannot find the square root of 18 here. Hence, the method is not applicable.

Square Roots of Numbers

• Square root of 28 = 5.29150262213
• Square root of 3 = 1.73205080757
• Square root of 6 = 2.44948974278
• Square root of 7 = 2.64575131106
• Square root of 8 = 2.82842712475
• Square root of 9 = 3
• Square root of 40 = 6.32455532034
• Square root of 49 = 7
• Square root of 180 = 13.416407865

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Solved Examples

Q.1: Find the square root of 100/18.

Solution: Given, 100/18

Suppose the square root of 100/18 is x

X = √(100/18)

X = √100/√18

√100 = 10

√18 = 3√2

X = 10/(3√2)

We can further rationalize the denominator.

Q.2: Find the area of a chessboard whose side is equal to 3√2 cm, each.

Solution: Given, the side of the chessboard is 3√2 cm.

As we know, the chessboard is in square shape. Therefore,

Area of the chessboard = (length of the side)²

Area = (3√2)²

Area = 18 sq.cm.

Therefore, the area of the chessboard is 18 sq.cm.

Q.3: What is the value of (√18)³?

Solution: The value of (√18)³ is:

(√18)³ = (3√2)³

= 3³ x (√2)³

= 27 x 2√2

= 54√2

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