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How do you simplify 4 square roots of 125?

Last updated date: 21st Feb 2024
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IVSAT 2024
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Hint: In order to solve this question, we are required to simplify the given number consisting of a square root. Thus, we will express the given square root number as a product of its factors using the prime factorization method. Then simplifying the square root of 125 and multiplying it by 4, we will get the required answer.

Complete step-by-step answer:
In order to simplify \[4\sqrt {125} \]
First, we have to find the square root of 125 using the prime factorization method.
Therefore, prime factorization of 125 is.
We can see that 125 is an odd number and 2 and 3 are not a factor of 125.
So, we will divide 125 by the least prime number 5. Therefore, we get
\[125 \div 5 = 25\]
Dividing 125 by 5, we get
\[25 \div 5 = 5\]
As we have obtained the quotient as a prime number, so we will not divide the number further.
Hence, 125 can be written as:
\[125 = 5 \times 5 \times 5\]
But, we will take out 1 factor from the pair of common factors in order to find the square root.
Hence, taking square root on both sides, we get,
\[\sqrt {125} = 5\sqrt 5 \]
Therefore, we can write:
 \[4\sqrt {125} = 4 \times 5\sqrt 5 \]
Multiplying the terms, we get
\[ \Rightarrow 4\sqrt {125} = 20\sqrt 5 \]

Hence, 4 square roots of 125 can be simplified as 20 square roots of 5.
Thus, this is the required answer.

Note: We know that prime factors are those factors that are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself. In order to express the given number as a product of its prime factors, we are required to do the prime factorization of the given number. Factorization is a method of writing an original number as the product of its various factors. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.