Question

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 3 e^{2 t} $$

Short answer

Question: Rewrite the given expression as a power raised to a power: $$3 e^{2 t}$$. Answer: The expression can be written as $$(3e^2)^t$$.

Step-by-step solution

Identify the base and exponent

The expression is given as $$3 e^{2 t}$$. Here, the base is $$e$$ (the natural exponential base) and the exponent is $$2t$$.

Write the expression as a power raised to a power

To write the expression as a power raised to a power, we will use the properties of exponents, namely the power rule, which states that $$(a^m)^n = a^{m \cdot n}$$. Therefore, the expression can be written as: $$(3e^2)^t$$, which represents the power raised to a power form.

Key concepts

Base and Exponent

In mathematics, when you see an expression like \(3 e^{2t}\), you're looking at what's known as an exponential expression. In this type of expression, there are two main components to understand: the base and the exponent.

The **base** is the number or expression that is being multiplied. In the expression \(3 e^{2t}\), the base is \(e\), known as the natural exponential base, often used in calculations involving growth and decay. It's a special number approximately equal to 2.71828.

The **exponent** indicates how many times the base is multiplied by itself. In \(3 e^{2t}\), the exponent is \(2t\). This means \(e\) is raised to the power of \(2t\), and represents repeated multiplication of \(e\) a total of \(2t\) times.

Mastering the concept of base and exponent is crucial for understanding more complex exponential expressions and how they can be manipulated.

Power Raised to a Power

A key skill when working with exponents is re-writing expressions using the power rule, especially in simplifying exponential expressions as a power raised to a power. This involves recognizing situations where an exponent itself is an exponential expression.

The power rule you're applying when rewriting expressions states that \((a^m)^n = a^{m \cdot n}\). This is especially useful when you need to perform additional operations with exponential expressions.

Consider the expression \(3 e^{2t}\). To write this differently, you could express it as \((3e^2)^t\). Here, you've used the power rule by recognizing that \(e^{2t}\) can be seen as \((e^2)^t\). This transforms the expression into a form that shows the power raised to another power, making it easier to handle in future operations. Think of it like stacking exponents, where exponentiation is performed more than once on the same base.

Properties of Exponents

The properties of exponents are essential tools that help manipulate and simplify exponential expressions. They can turn complex calculations into more manageable and understandable steps.

Some fundamental properties you might encounter are:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product Rule: \((ab)^m = a^m \cdot b^m\)

In the exercise, the Power of a Power Rule was employed to transform the expression from \(e^{2t}\) to \((e^2)^t\). This rule is useful for breaking down complex exponents by allowing you to multiply the powers, simplifying expressions when dealing with multiple layers of exponentiation.

Understanding these properties not only helps in re-writing expressions but also in solving exponential equations and comparing exponential terms. They are indispensable tools in many areas of mathematics, allowing for flexibility and simplification in otherwise complicated problems.