The square root of 112 can be determined by the prime factorisation and long division method. As 112 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 112 is denoted as √112, and in the exponential form, it is written as (112)½. In this article, we shall learn how to find the square root of 112.
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Square Root of 112 |
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Square of 112 |
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What Is the Square Root of 112?
The square root of 112 is a number whose square gives the result 112. Now, 112 is not a perfect square number because we cannot find any integer which could be multiplied twice to get 112. Thus, we find an approximate value of the square root of 112, as it is an irrational number.
The square root of 112 can also be determined by finding the roots of the quadratic equation
x2 – 112 = 0
⇒ x2 = 112
Taking square roots on both sides, we get,
⇒ x = √112
⇒ x = ±4√7
How to Find the Square Root of 112?
Let us find the square root of 112 using the prime factorisation and long division method. The repeated subtraction method will not work as 112 is not a perfect square, and its square root is an irrational number.
Prime Factorisation Method
To find the square root of 112 by the prime factorisation method, we first prime factorise 112 and then make pairs of two to get the square root.
Prime factorisation of 112 = 2 × 2 × 2 × 2 × 7
Square root of 112 = √[2 × 2 × 2 × 2 × 7] = 2 × 2 × √7 = 4√7
Long Division Method
To find the square root of 112 by the long division method, we shall write 112 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 112
Example 1:
Find the semi-perimeter of the circle whose area is 112𝜋 m2.
Solution:
Let r be the radius of the circle.
The area of the circle = 𝜋r2 = 112𝜋 m2
⇒ r2 = 112 (taking square root on both the sides)
⇒ r = √112
⇒ r = 4√7 m (taking the positive root as length cannot be negative)
∴ the semi-perimeter of the circle = 𝜋r = 22/7 × 4√7 = 88/√7 m
Example 2:
Find the length of the base of a triangle whose area is 28 cm2 and the height is half its base.
Solution:
Let b be the base of the triangle.
Height of the triangle = b/2
Area of the triangle = ½ × b × b/2 = 28 cm2
⇒ ¼ × b2 = 28
⇒ b2 = 28 × 4 = 112 (taking square root on both sides)
⇒ b = √112 = 4√7 cm
∴ the length of the base of the triangle is 4√7 cm.
Example 3:
What is the smallest number that should be subtracted from 112 to make it a perfect square number? Also, find the square root of the number obtained.
Solution:
Clearly, 100 = 102 < 112
Therefore, we must subtract 12 from 112 to make it a perfect square number. 112 – 12 = 100 and
√100 = ± 10
Frequently Asked Questions on Square Root of 112
What is the square root of 112?
The square root of 112 is 4√7 or 10.58 (approx.).
Is 112 a perfect square number?
No, 112 is not a perfect square number, as it cannot be expressed as the square of any integer.
Is the square root of 112 rational or irrational?
The square root of 112 is an irrational number.
What is the prime factorisation of 112?
The prime factorisation of 112 is 2 × 2 × 2 × 2 × 7.
Is the square root of 112 a real number?
Yes, the square root of 112 is a real number.
What is the cube root of 112?
The cube root of 112 is 4.82 (approx.).
What number should be divided from 112 to get the quotient a perfect square number?
By the prime factorisation of 112 = 2 × 2 × 2 × 2 × 7, only 7 is left unpaired. Hence, we must divide 7 from 112. Therefore, 112 ÷ 7 = 16 is a perfect square number.
