The square root of 99 can be determined by the prime factorisation and long division method. As 99 is not a perfect square number, we get an irrational root. A perfect square number is a number that can be written as a square of an integer. The square root of a number is just the opposite of squaring. In radical form, the square root of 99 is denoted as √99, and in the exponential form, it is written as (99)½. In this article, we shall learn how to find the square root of 99.
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What Is the Square Root of 99?
The square root of 99 is a number whose square gives the result 99. Now, 99 is not a perfect square number because we cannot find any integer which could be multiplied twice to get 99. Thus, we find an approximate value of the square root of 99, as it is an irrational number.
The square root of 99 can also be determined by finding the roots of the quadratic equation
x2 – 99 = 0
⇒ x2 = 99
Taking square roots on both sides, we get,
⇒ x = √99
⇒ x = ±3√11
How to Find the Square Root of 99?
Let us find the square root of 99 using the prime factorisation and long division method. The repeated subtraction method will not work as 99 is not a perfect square, and its square root is an irrational number.
Prime Factorisation Method
To find the square root of 99 by the prime factorisation method, we first prime factorise 99 and then make pairs of two to get the square root.
Prime factorisation of 99 = 3 × 3 × 11
Square root of 99 = √[3 × 3 × 11] = 3 × 3 × √11 = 3√11
Long Division Method
To find the square root of 99 by the long division method, we shall write 99 as the dividend and pair its digits from right to left. Now, we shall calculate the square root as follows:
To learn how to find the square root of any number by the long division method, click here.
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Solved Examples on Square Root of 99
Example 1:
Find the diameter of the circle whose area is 99𝜋 m2.
Solution:
Let r be the radius of the circle.
The area of the circle = 𝜋r2 = 99𝜋 m2
⇒ r2 = 99 (taking square root on both sides)
⇒ r = √99
⇒ r = 3√11 m (taking the positive root as length cannot be negative)
∴ the diameter of the circle = 2 × 3√11 = 6√11 m
Example 2:
Find the length of the base of a triangle whose area is 33 m2 and the height is two-third its base.
Solution:
Let b be the base of the triangle.
Height of the triangle = 2b/3
Area of the triangle = ½ × b × 2b/3 = 33 m2
⇒ ⅓ × b2 = 33
⇒ b2 = 99 (taking square root on both sides)
⇒ b = √99 = 3√11 m
∴ the length of the base of the triangle is 3√11 m.
Example 3:
What is the smallest number that should be added to 99 to make it a perfect square number? Also, find the square root of the number obtained.
Solution:
Clearly, 99 < 100 = 102
Therefore, we must add 1 with 99 to make it a perfect square number.
√100 = ± 10.
Frequently Asked Questions on Square Root of 99
What is the square root of 99?
The square root of 99 is 3√11 or 9.94 (approx.).
Is 99 a perfect square number?
No, 99 is not a perfect square number, as it cannot be expressed as the square of any integer.
Is the square root of 99 rational or irrational?
The square root of 99 is an irrational number.
What is the prime factorisation of 99?
The prime factorisation of 99 is 3 × 3 × 11.
Is the square root of 99 a real number?
Yes, the square root of 99 is a real number.
What is the cube root of 99?
The cube root of 99 is 4.626 (approx.).
What number should be divided from 99 to get the quotient a perfect square number?
By the prime factorisation of 99 = 3 × 3 × 11, only 11 is left unpaired. Hence, we must divide 99 by 11. Therefore, 99 ÷ 11 = 9 is a perfect square number.
