Final answer:
Both expressions √13^2 and (√13)^2 result in the original number 13, because the square (squaring) and the square root operations are inverses of each other and cancel out.
Step-by-step explanation:
The expression √13^2 simplified means taking the square root of 13 and then squaring the result. Mathematically, these operations are inverses of each other, meaning that the square (squaring) and the square root operations cancel each other out, yielding the original number underneath the radical, which is 13. To illustrate this with another example, (√5)^2 would simply be 5. The confusion often arises because of the order and interpretation of operations, but remember that raising a square root to the power of 2 'undoes' the square root.
On the other hand, the expression (√13)^2 demonstrates the same principle but is written with parentheses to clearly show the order of operations: first, the number 13 is under the square root, and then the result of that root is squared. This will also result in the original number, 13, as the square root and the square operation nullify each other.
Understanding the exponent rules, such as when we raise a base number with an exponent to another power (as per Eq. A.8), we multiply the exponents. However, since the square root can be represented as a fractional exponent (that is, √x = x^(1/2)), squaring the square root ((√x)^2 = x^(1/2 * 2)) simplifies to x^1, which is just x. This concept helps us see why √13^2 = 13 and why (√13)^2 = 13 as well.