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How do you simplify $ 4 $ square root $ 27 $ $ - $ square root $ 75? $

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Last updated date: 20th Jul 2024
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Answer
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Hint: As we know that square root can be defined as a number which when multiplied by itself gives a number as the product. For example $ 5 \times 5 = 25 $ , here the square root of $ 25 $ is $ 5 $ . There is no such formula to calculate square root formula but two ways are generally considered. They are the prime factorization method and division method. The symbol $ \sqrt {} $ is used to denote square root and this symbol of square roots is also known as radical.

Complete step-by-step answer:
Here we have to find the value of $ 4\sqrt {27} - \sqrt {75} $ , since both are non perfect squares so we will factorise it under the root: $ \sqrt {27} $ can be written as $ 9 \times 3 $ and we know that $ 9 $ is a perfect square and we can take out of the radical so we get,
$ \sqrt {{3^2}} \times \sqrt 3 $ $ = 3\sqrt 3 $ .
Similarly $ \sqrt {75} $ can be further written as
$ \sqrt {75} \Rightarrow {5^2} \times 3 $ , It gives $ \sqrt {{5^2}} .\sqrt 3 \Rightarrow 5\sqrt 3 $ .
By putting the values together we have:
$ 4\sqrt {27} - \sqrt {75} = 4 \times 3\sqrt 3 - 5\sqrt 3 $ .
On multiplying we have $ 12\sqrt 3 - 5\sqrt 3 $ . It gives the value $ 7\sqrt 3 $
Hence the answer is $ 7\sqrt 3 . $ .
So, the correct answer is “ $ 7\sqrt 3 $ ”.

Note: The above given numbers are non-perfect squares as we know that a non-perfect square is a number that there is no rational number i.e. it is considered as an irrational number. Their decimal does not end and they do not repeat a pattern so they are also non-terminating and non-repeating numbers. The number written inside the square root symbol or radical is known as radicand. We know that all real numbers have two square roots, one is a positive square root and another one is a negative square root. The positive square root is also referred to as the principal square root.