91Ó°ÊÓ

When simplifying expressions involving exponents, the product rule is an essential tool. This rule states that when you multiply terms with the same base, you add the exponents. This is because the operation of raising to powers involves repeated multiplication. For example, consider multiplying \(a^5\), \(a^2\), and \(a^7\). All of these terms have the base \(a\). According to the product rule, you add the exponents together:

• First term: \(a^5\)
• Second term: \(a^2\)
• Third term: \(a^7\)

To simplify \(a^5 \times a^2 \times a^7\), add their exponents: \[a^{5+2+7} = a^{14}\] This transformation shows how exponential properties make complex multiplications simpler and how like bases can be streamlined by summing their powers.

Multiplying coefficients occurs in expressions where both numbers and variables are present. Coefficients are the numerical parts of terms, and it is important to handle these separately from the variables with exponents. This is straightforward—just multiply the numbers using basic arithmetic. In the given expression, the coefficients are \(\frac{1}{6}\), \(-3\), and \(4\). To find the product:

• Multiply \(\frac{1}{6} \times -3 = -\frac{1}{2}\)
• Then, multiply the result by \(4\): \(-\frac{1}{2} \times 4 = -2\)

This yields \(-2\), which represents the product of the numerical parts alone. It's crucial to distinguish between coefficients and variables so as not to confuse arithmetic multiplication with the rule for exponents.

Variables with powers can appear daunting, but they are simpler to manage with the right rules. A variable with an exponent is a variable raised to a power, illustrating how many times to use the variable as a factor in multiplication. When you have an expression like \(a^5 \times a^2 \times a^7\), the powers of the variable \(a\) tell you the number of times \(a\) is multiplied by itself. The exponents 5, 2, and 7 represent these multiplicative actions individually:

• The base is \(a\).
• Individual powers are \(5, 2, and 7\).

These exponents are added together by the product rule, showing how cumulative multiplications build from the base. The final combined power—\(a^{14}\)—demonstrates the effective multiplication of the variable \(a\) 14 times. Recognizing the powers' role helps transform complex algebraic expressions into simpler forms.