Answer

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**Hint:**First we have to define the terms we need to solve the problem.

Let the first term of the equation be $a$

Let the second term of the equation be $b$

Subtraction of the squares of both terms is given by the product of sum of both terms and difference of both terms.

Formula Used: ${a^2} - {b^2} = (a - b)(a + b)$

**Complete step by step answer:**

We need to find the value of\[{513^2} - {512^2}\]

Let us consider $a$ be the first term.

Thus,

$a = 513$

Let us consider $b$ is the second term.

Thus,

$b = 512$

To find the required value use the formula,

${a^2} - {b^2} = (a - b)(a + b)$

Let us put the values of $a$ and $b$ in the above formula,

${513^2} - {512^2} = (513 - 512)(513 + 512)$

Let us solve the above equation,

On performing the subtraction in first bracket we get,

${513^2} - {512^2} = (1)(513 + 512)$

On performing the addition in second bracket we get,

${513^2} - {512^2} = (1)(1025)$

On performing the multiplication of the both brackets we get,

${513^2} - {512^2} = 1025$

This is the required value.

**Note:**The identity ${a^2} - {b^2} = (a - b)(a + b)$is the simplest standard identity. This identity can be proved geometrically. In the formula mentioned above, $(a + b)(a - b)$are the factors of${a^2} - {b^2}$. These problems look difficult but the values are given such that factors give very simple values, which can be multiplied easily and answers can be calculated. If values of your factors calculated are complicated, check your calculation again and make sure you calculate the correct values. The value of at least one of the factors is mostly very simple and easy to multiply with other factors. If in such questions, this identity is not used, then the calculations become very complicated and result in huge numbers.

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