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Exponents provide a way to express repeated multiplication of the same number. They are found in expressions like \( t^6 \), where \( t \) is multiplied by itself six times. The general form is \( a^n \), where \( a \) is called the base and \( n \) is the exponent. This means that the base is multiplied by itself "n" times. For instance, \( 2^3 \) means \( 2 \times 2 \times 2 \), which equals 8. When multiplying expressions with the same base but different exponents, you use the power rule of exponents. This allows you to add the exponents together while keeping the base the same:

• \( a^m \times a^n = a^{m+n} \)

In the given problem, since both terms use the base \( t \), you simply add the exponents: \( t^6 \times t^5 = t^{11} \). This simplifies the process, making multiplication more efficient.

Coefficients are numbers that multiply a variable. In the algebraic expression \( 9t^6 \), 9 is the coefficient. They dictate how many times you multiply the base expression, often setting the magnitude for terms. In multiplication, you deal with coefficients by multiplying them just like any other numbers. In the exercise, the coefficients are 9 and -3, so you calculate \( 9 \times (-3) \).Calculating it:

• 9 \( \times \) (-3) = -27

This results in a new coefficient for the expression after algebraic multiplication. Even when dealing with complex variables, handling coefficients remains straightforward: simply multiply them, regardless of any exponent rules applied to the variables.

The power rule is a fundamental concept in algebra that simplifies the multiplication of powers with the same base. This rule states that when multiplying two powers with the same base, you simply add the exponents. Formally, it's written as \( a^m \times a^n = a^{m+n} \). Here’s why it's important:

• It allows for quicker calculations in algebra and beyond.
• It keeps expressions tidy by avoiding expanding and manually multiplying terms.

In your problem, made simple by this rule, when faced with \( t^6 \times t^5 \), you apply it directly to gain \( t^{11} \). This saves time and reduces error, especially useful in more complex algebraic manipulations.
Mastering the power rule helps greatly in math, ensuring you can quickly and correctly handle expressions with exponents.