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Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In expressions like \(x^n\), \(x\) represents the base and \(n\) the exponent. This operation means that you multiply the base, \(x\), by itself \(n\) times. For example, \(3^4\) (read as "three to the fourth power") means multiplying 3 by itself 3 more times: \(3 \times 3 \times 3 \times 3 = 81\). Such powers of a number simplify expressions involving repeated multiplication and allow quick calculation of large numbers.

When dealing with expressions like \(-7^2\), it's crucial to understand the exponent only applies to the base immediately next to it unless there are parentheses. Thus, \(7^2\) becomes 49, but because \(-7^2\) is interpreted as \(-(7^2)\), the result is -49. This subtlety often leads to confusion, so always watch out for placement of negative signs and parentheses.

Negative exponents indicate the reciprocal of the base raised to the positive opposite of the exponent. For instance, \(x^{-n}\) means \(1/x^n\). This changes the magnitude (size) without affecting the sign of the base itself. Thus, if \(x = 5\) and \(n = -2\), then \(x^{-2} \) becomes \(1/5^2 = 1/25\).

It's important to differentiate between negative signs in the base and negative exponents. A negative sign in the base, like \(-4^{-3}\), doesn't change how you handle the exponent, but \(-4\) is still the base. Using a negative exponent flips the fraction: \(-1/64\) since \(-4^{-3} = -(1/4^3) = -(1/64)\). This concept helps in balancing expressions, especially when needing to simplify complex fractions or make equations easier to handle.

The order of operations is a set of rules mathematicians have agreed upon to avoid confusion in expressions requiring multiple operations. The acronym PEMDAS helps us remember the sequence:

• P is for Parentheses – do these first.
• E is for Exponents – handle these second.
• MD from Multiplication and Division – go from left to right, whichever comes first.
• AS for Addition and Subtraction – proceed similarly from left to right.

Following PEMDAS, if we encounter an expression like \(-7^{2} + 5^{2}\), the order dictates exponentiation first. Calculate \(7^2 = 49\) and then \(5^2 = 25\). Next, apply multiplication or division if any existed but here, we subtract since we treat the negative as \(-(7^2) = -49\). Lastly, add \(-49 + 25 = -24\). Adhering to this sequence ensures consistency in solving expressions and preventing costly mistakes.