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How do you simplify \[\sqrt {567} \]?

Last updated date: 20th Jun 2024
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Hint: Square root of a number is a value, which on multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as \[x = \sqrt y \] or we can express the same equation as \[{x^2} = y\]. Here we can see that 294 is not a perfect square. To solve this we factorize the given number.

Complete step-by-step solution:
\[\sqrt {567} \]
567 can be factorized as,
\[567 = 1 \times 3 \times 3 \times 3 \times 3 \times 7\]
We can see that 3 is multiplied twice two times, we multiply that we get,
\[567 = {3^2} \times {3^2} \times 7\].
\[ \Rightarrow \sqrt {567} = \sqrt {{3^2} \times {3^2} \times 7} \]
Now we know that square and square root will get cancel and we take term out the radical symbol,
\[ = 3 \times 3\sqrt 7 \]
\[ = 9\sqrt 7 \]. This is the exact form. We can stop here.
We also put it in the decimal form.
We know that \[\sqrt 7 = 2.645\] and multiplying this with 9 we get,
\[ = 9 \times 2.645\]
\[ = 23.81\]. This is the decimal form.

Note: Here \[\sqrt {} \] is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.