Question

In Exercises 3–12, simplify the expression. $$ e^3 \cdot e^5 $$

Short answer

The simplified expression is \(e^8\).

Step-by-step solution

Identifying the base and exponents

In this problem, the base for both of the expressions is the constant \(e\). Meanwhile, the first exponent is 3, and the second exponent is 5.

Applying the rules of exponents

When we're multiplying two exponential expressions that have the same base, we can combine the two expressions into one by adding the exponents. This means that we can rewrite \(e^3 \cdot e^5\) as \(e^{3+5}\).

Simplifying the exponents

Now, all we need to do is add the exponents together. This gives us \(e^{3+5} = e^8\).

Key concepts

Exponents

Exponents, also known as powers, are a shorthand way of expressing repeated multiplication. For example, when you see an expression like \(2^3\), this tells us to multiply 2 by itself three times: \(2 \times 2 \times 2\). The number 2 is the base, and the number 3 is the exponent. The role of the exponent is crucial; it determines how many times the base is used as a factor in the multiplication.

Understanding exponents is fundamental in mathematics, as they are used to represent very large or very small numbers succinctly, and occur in a wide range of mathematical contexts, including algebraic equations and scientific notation.

Rules of Exponents

Multiplying with the Same Base

When multiplying exponential expressions with the same base, you add the exponents. For the exercise \(e^3 \cdot e^5\), you combine the exponents by performing the operation \(3 + 5\) to get \(e^8\). It's essential to remember that this rule only applies if the bases are the same.

Power to a Power

When you raise an exponent to another exponent, you multiply the exponents. For instance, \( (2^3)^2\) becomes \(2^{3\cdot2}\) or \(2^6\).

Division with the Same Base

When dividing exponential expressions with the same base, you subtract the exponents. If the exercise were \(e^5 / e^3\), the solution would be \(e^{5-3}\), which simplifies to \(e^2\).

These rules help simplify expressions, solve algebraic equations, and understand the growth patterns in exponential functions.

Exponential Functions

Exponential functions are mathematical functions in the form \(f(x) = a \cdot b^x\), with the base \(b\) being a positive real number not equal to 1, and \(x\) is the exponent. These functions show up extensively in various contexts such as compound interest, population growth, and radioactive decay.

The function describing an exponential growth or decay depends on the base \(b\): if \(b > 1\), the function grows as \(x\) increases, and if \(0 < b < 1\), it represents decay. The constant \(e\), approximately 2.71828, is a special base that often appears in natural processes. The previously solved exercise \(e^3 \cdot e^5 = e^8\) is a simplified version of an exponential function where the exponential rules are applied to simplify the expression efficiently.