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Negative numbers are numbers that are less than zero. They are used to represent quantities that are less than none or a lack of something. You can think of negative numbers as the opposite of positive numbers.
When dealing with negative numbers in expressions, keep these tips in mind:

• Multiplying two negative numbers results in a positive number (e.g., \((-1) imes (-1) = 1\)).


• Multiplying a negative number with a positive one results in a negative number (e.g., \((-1) imes 3 = -3\)).


• The negative of a negative number is a positive number (e.g., the negative of \(-5\) is \(+5\)).

When raising negative numbers to powers, pay attention to the exponent. If the exponent is odd, the result will be negative. If it's even, the result will be positive. For example, \((-1)^3 = -1\) (because 3 is odd) and \((-2)^4 = 16\) (because 4 is even).
This concept is crucial for solving problems where negative numbers are raised to a power.

Fraction simplification involves reducing a fraction to its simplest form. In other words, simplifying a fraction means rewriting it in a way that both the numerator and the denominator have no common factors other than 1.
To simplify fractions, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.


• Divide both the numerator and the denominator by their GCD.

For example, to simplify \(\frac{8}{12}\), notice that 4 is the largest number that can divide both 8 and 12. Therefore, divide the numerator and the denominator by 4 resulting in \(\frac{2}{3}\).
In our exercise, after calculating powers, the fraction was \(\frac{-81}{81}\). Both numbers share 81 as a common factor, so dividing the numerator and denominator by 81 gives \(-1\). This shows that understanding fraction simplification is critical in solving algebraic expressions effectively.

Algebraic expressions involve numbers, variables, and operations. They are the building blocks of algebra, allowing us to model real-world situations mathematically.
An algebraic expression could include more complex mathematical operations like addition, subtraction, multiplication, division, and exponentiation.
To understand and work with algebraic expressions:

• Simplify complex expressions by following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction, commonly known as PEMDAS).


• Combine like terms, which are the terms with the same variable raised to the same power.


• Manipulate negative signs carefully, especially when they appear with powers or within parentheses.

In exercises involving expressions like \(\frac{-1^{3} \cdot(-3)^{4}}{9^{2}}\), it’s important to manage elements individually, calculate exponents first, and apply the properties of negative numbers, before simplifying the entire expression. Despite their complexity, algebraic expressions can be simplified step-by-step with a solid understanding of these basics.