Expanding the Expression: a^2z(t - a)\left(\dfrac{1}{z} + \dfrac{1}{a} \right)

As a skilled high school mathematics teacher, I'm excited to guide you through the process of expanding the given algebraic expression: a^2z(t - a)\left(\dfrac{1}{z} + \dfrac{1}{a} \right).

Expanding an expression means to multiply all the terms together, resulting in a single expression with no parentheses, fractions, or exponents. This is an essential skill in high school mathematics, as it allows you to simplify complex expressions and prepare them for further algebraic manipulations.

Let's break down the step-by-step process:

1. Identify the individual terms: The expression consists of the following terms:

• a^2
• z
• (t - a)
• (1/z + 1/a)

1. Multiply the terms: To expand the expression, we need to multiply all the terms together. We can do this by using the distributive property:

   a^2z(t - a)\left(\dfrac{1}{z} + \dfrac{1}{a} \right) = a^2z(t - a)\left(\dfrac{1}{z}\right) + a^2z(t - a)\left(\dfrac{1}{a}\right) = a^2zt - a^3z + a^2zt - a^3z = 2a^2zt - 2a^3z

2. Simplify the expression: The final expanded expression is: 2a^2zt - 2a^3z

In this expanded form, the expression is now ready for further algebraic manipulations, such as factoring, solving equations, or graphing.

Remember, the key to expanding expressions is to carefully apply the distributive property and multiply each term in the parentheses by each term outside the parentheses. With practice, this process will become second nature, and you'll be able to tackle more complex algebraic expressions with confidence.

If you have any further questions or need additional guidance, feel free to reach out to me as your excellent high school mathematics teacher. I'm always happy to provide more detailed explanations and support to help you excel in your mathematical studies.