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Exponents are a way to express repeated multiplication of the same number or variable. The rules of exponents help simplify expressions and make calculations easier. When multiplying terms that have the same base, you simply add the exponents together. This is known as the product of powers property.

For example, if you have two expressions like \( x^a \) and \( x^b \), and you multiply them together, the result is \( x^{a+b} \). This is because when you're multiplying, you are essentially adding the number of times the base is used as a factor. Let's consider specific examples to clarify this rule:

• \( x^3 \times x^2 = x^{3+2} = x^5 \)
• \( y^4 \times y^6 = y^{4+6} = y^{10} \)

This rule helps in dealing with more complex expressions such as \( x^4 \times x^7 = x^{11} \) and \( y \times y^{11} = y^{12} \). Simplifying expressions using these exponent rules is a crucial skill in algebra and beyond.

When you are faced with an expression that involves multiplying variables, it's essential to handle each variable separately while adhering to the algebraic rules. If variables have different bases, they are just multiplied as they are. If they have the same base, you'll apply the rule of exponents.

In an expression like \( x^4 \times x^7 \), since both terms have the same base \( x \), you add the exponents (4 and 7) to get \( x^{11} \). Similarly, for \( y \times y^{11} \), we assume \( y \) is \( y^1 \) and thus get \( y^{1+11} = y^{12} \). Handling each variable separately helps in clearer understanding and avoids confusion, especially in larger expressions.

Numeric multiplication is often the first step when dealing with expressions, as it provides a simplified number to work with alongside the variables. It follows the basic arithmetic rules.

In the given expression \((-5) \cdot (-6)\), we first multiply the numeric coefficients. Multiplying two negative numbers results in a positive product, so \(-5 \times -6 = 30\). This process simplifies the expression significantly before combining it with the variable results.

• Multiplication of negatives: The product of two negative numbers is always positive.
• Product operations: Simply multiply as usual, keeping track of positive and negative signs.

Mastering numeric multiplication is key to solving more complex algebraic problems that involve variables.