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Exponents are a shorthand way to express repeated multiplication of a number or variable. When you see an expression like \(a^5\), it means that the number \(a\) is multiplied by itself 5 times.

There are some key rules to keep in mind when working with exponents:

• **Product of Powers Rule:** When you multiply two expressions that have the same base, like \(a^m \times a^n\), you simply add the exponents. The result is \(a^{m+n}\).
• **Power of a Power Rule:** When you raise an exponent to another exponent, like \((a^m)^n\), you multiply the exponents. The result is \(a^{m \times n}\).
• **Zero Exponent Rule:** Any non-zero number raised to the power of zero is 1, such as \(a^0 = 1\).

In the original exercise, we applied these rules to simplify the expression by distributing the exponent 2 over the terms inside the parentheses, \((9a^5b^6)^2\), resulting in \(81a^{10}b^{12}\).

When you multiply expressions, you usually deal with multiplying coefficients (the numbers) and variables (the letters with exponents) separately. This approach makes complicated algebraic expressions more manageable.

The steps to multiply algebraic expressions are:

• Multiply the coefficients (the numerical parts).
• For each variable, use the product of powers rule to add the exponents if the base is the same.

For example, in the solved exercise, the expression \(81a^{10}b^{12}\) was multiplied with \(8a^5b\). This required multiplying 81 by 8 to get 648. Then, the product of powers rule was used for the variables: add the exponents of \(a\) (\(a^{10+5} = a^{15}\)) and \(b\) (\(b^{12+1} = b^{13}\)).

This combination of multiplying coefficients and manipulating variables ensures that expressions are simplified correctly.

The distributive property is a foundational concept in algebra that allows you to multiply a single term by each term within a set of parentheses. It states that for any numbers \(a, b,\) and \(c\), \(a(b+c) = ab + ac\). This principle can be extended to multiplication of terms raised to an exponent.

In the exercise, we specifically used the concept of distributing the exponent over a parenthetical expression. Applying the exponent \(n\) to an expression \((p \cdot q)^n\) means you raise both \(p\) and \(q\) to the power of \(n\) separately, resulting in \(p^n \cdot q^n\).

In our original problem, distributing the exponent 2 over \((9a^5b^6)\) gave us \((9^2)(a^{5 \times 2})(b^{6 \times 2})\), simplifying to \(81a^{10}b^{12}\). The distributive property ensures that we account for all parts of the expression uniformly, leading to an accurate simplification.