Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ e^{t-1}(t+1) $$

Short answer

Question: Rewrite the expression \(e^{t-1}(t+1)\) as a product or a quotient. Answer: The expression can be rewritten as \(\frac{e^t}{e} \cdot (t+1)\).

Step-by-step solution

Identify terms

The given expression contains two terms: the exponential term (\(e^{t-1}\)) and the algebraic term (\((t+1)\)).

Rewrite as a product of exponentials

To rewrite the expression as a product or a quotient, we'll focus on the exponential term. Using the properties of exponentials, we can rewrite this term as follows: $$ e^{t-1} = e^t \cdot e^{-1} $$

Combine with the algebraic term

Now, we can combine the exponential component with the algebraic term \((t+1)\) to rewrite our original expression as a product: $$ e^{t-1}(t+1) = e^t \cdot e^{-1} \cdot (t+1) $$

Simplify the expression

The expression as it is currently, \(e^t \cdot e^{-1} \cdot (t+1)\), is already in a product form. However, it's useful to rewrite \(e^{-1}\) as a reciprocal to obtain a quotient: $$ e^{t-1}(t+1) = \frac{e^t}{e} \cdot (t+1) $$ Now the expression is written as both a product and a quotient, and the solution is complete.

Key concepts

Algebraic Expressions

Algebraic expressions are a combination of constants, variables, and mathematical operations (like addition, subtraction, multiplication, and division) grouped together. These expressions are fundamental building blocks in algebra. They allow us to represent problems and solve equations in a generalized form.

An expression like \(e^{t-1}(t+1)\) is considered algebraic because it includes an exponential term \(e^{t-1}\) and an algebraic term \((t+1)\), which are both joined by multiplication. Here, \(t\) is a variable, and the overall expression can change based on the value of \(t\).

This expression requires us to apply algebraic operations to rewrite it in different forms, such as a product or a quotient. Understanding algebraic expressions' structure is crucial for manipulating and simplifying them, which is a key skill in mastering algebra.

Product and Quotient

When dealing with expressions, it's often useful to rewrite them in terms of simpler or more familiar operations like products and quotients. Rewriting an expression as a product involves expressing it entirely as a multiplication of terms.

In the given exercise, the expression \(e^{t-1}(t+1)\) is simplified to a product as shown: \(e^t \cdot e^{-1} \cdot (t+1)\). Here, each component of the original expression is multiplied together.

• Product Form: Simply means firmly expressing through multiplication.
• Quotient Form: Involves division, such as converting \(e^{-1}\) into the form \(\frac{1}{e}\).

Writing an expression as a quotient, particularly when dealing with negative exponents, can provide additional insights into the relationship between the terms. It reflects that part of the multiplication involves "dividing" by a particular component.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with expressions that include exponential terms. These properties allow us to manipulate and transform expressions efficiently.

The fundamental properties involve:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero Exponent Rule: \(a^0 = 1\), for any \(a eq 0\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

In our expression, \(e^{t-1} \) becomes \(e^t \cdot e^{-1} \). The property that \(a^{-n} = \frac{1}{a^n}\) transforms \(e^{-1}\) to \(\frac{1}{e}\). This ability to change forms using exponent rules is essential in algebra, enabling us to analyze and solve for variables more straightforwardly. Mastering these rules lets you tackle various algebraic challenges with confidence.