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# Show that $500$ is not a perfect square?

Last updated date: 25th Jul 2024
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Hint: In the given question, we are required to show that number provided to us, $500$ is not a perfect square. So, in order to show that the given number is not a perfect square, we will try to compute the square root of the value. Square root of a number is a value, which when 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 ${x^2} = y$ .

We have to show that the number $500$ is not a perfect square. So, we try to find the square root of the number. If in any case, the square root of the number comes out to be a positive integer, then the number is a perfect square. Otherwise, the number is not a perfect square.
$500$ can be factorized as,
$500 = 2 \times 2 \times 5 \times 5 \times 5$
Now, we can see that $2$ is multiplied twice, so we write in exponential form and raise $2$ to the power $2$ . Similarly, we can see that $5$ is multiplied thrice, so we write in exponential form and raise $5$ to the power $3$ . So, we get,
$\Rightarrow 500 = {2^2} \times {5^3}$
Now, $\sqrt {500} = \sqrt {{2^2} \times {5^3}}$
We know that ${5^3} = {5^2} \times 5$ . So, ${5^3}$ can be written as ${5^2} \times 5$
Now, $\sqrt {500} = \sqrt {{2^2} \times {5^2} \times 5}$
Since we know that ${2^2}$ and ${5^2}$ are perfect squares. So, we can take these outside of the square root. So, we have,
So, $\sqrt {500} = 2 \times 5 \times \sqrt 5$
Since $5$ is not a square number, we keep it inside the square root.
$\Rightarrow \sqrt {500} = 10\sqrt 5$
This is the simplified form of $\sqrt {500}$ .
Now, we see that the square root of the number $500$ is not a perfect square as the prime factors of the number are not in pairs. So, we can conclude that the number $500$ is not a perfect square.

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.