Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ e^{2+r} $$

Short answer

Question: Rewrite the expression $$e^{2+r}$$ as a product or a quotient using the properties of exponents. Answer: $$e^{2+r} = e^2 \cdot e^r$$

Step-by-step solution

Apply the exponent laws.

We apply the exponent law \(a^{x+y} =a^x \cdot a^y\) to the given expression $$e^{2+r}$$. Let a = e, x = 2, and y = r. Rewrite the expression as the product of two exponential terms with base e. $$ e^{2+r} = e^2 \cdot e^r $$ The final expression is now rewritten as a product: $$ e^{2+r} = e^2 \cdot e^r $$

Key concepts

Product of Exponents

When we talk about the "product of exponents," we're referring to a rule that helps us simplify expressions involving powers or exponents. Specifically, the rule states that when you multiply powers with the same base, you can add their exponents together. This principle is often used to simplify expressions and solve mathematical problems.

For example, imagine you have an expression like \( a^{m+n} \). According to the exponentiation rule, this can be written as \( a^m \cdot a^n \). The base "a" stays the same, and the exponents "m" and "n" are split into separate exponential terms that are multiplied together. This transformation is straightforward, and that's what makes it so useful.

This rule applies to any scenario where you encounter a single base raised to an exponent that is a sum, like \( e^{2+r} \), where "e" is the base, and "2" and "r" are the exponents. Using the product of exponents rule makes such expressions easier to manage.

Expressions

Expressions are combinations of numbers, variables, and operations like addition or multiplication. They're a way to represent a problem or a scenario in mathematics.

In the context of exponents, expressions can become quite complex when variables and numbers get raised to a power. For instance, an expression like \( e^{2+r} \) involves both a constant and a variable raised as exponents, which might seem confusing at first. But by applying exponent laws, we can break these down into simpler parts.

Breaking an expression containing exponents into simpler terms allows us to work with each part more easily. The process used in our example is to write the expression as a product, \( e^2 \cdot e^r \). Expressions can be represented as products or quotients, depending on their structure and the operations involved.

• Simplifying expressions helps in understanding and solving mathematical problems efficiently.
• Turning complex expressions into simpler parts by using laws like the product of exponents increases clarity.

Positive Variables

In mathematics, a variable is a symbol that represents a number, and when we say variables are positive, it means they take values greater than zero. This assumption simplifies many problems, particularly those involving powers and roots.

For an expression like \( e^{2+r} \), it's assumed that any variable, such as "r", is positive. This is an important consideration because negative values can affect the outcome significantly, creating different nuanced results. For instance, negative exponents indicate a reciprocal, like \( a^{-n} = \frac{1}{a^n} \).

Working with positive variables ensures that no unexpected transformations, such as flipping between multiplication and division, occur. This is especially beneficial when verifying that an expression remains manageable and results in predictable and straightforward outcomes. Ensuring variables are positive allows for consistent methods to simplify and solve problems using exponent laws effectively.

• Positive variables guarantee simpler and more predictable results.
• Assuming variables are positive helps maintain clarity and consistency in math problems.