Question

In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(e^{3}\right)\left(e^{4}\right)\) (b) \(\left(e^{3}\right)^{4}\) (c) \(\left(e^{3}\right)^{-2}\) (d) \(e^{0}\)

Short answer

(a) \(e^7\), (b) \(e^{12}\), (c) \(1/e^{6}\), (d) 1

Step-by-step solution

Solving Part (a)

This operation requires to add the exponents. Thus, \(\left(e^{3}\right)\left(e^{4}\right)= e^{3+4} = e^7\)

Solving Part (b)

This operation requires to multiply the exponents. Thus, \(\left(e^{3}\right)^{4}= e^{3*4} = e^{12}\)

Solving Part (c)

This operation requires using the negative exponent rule. So \(\left(e^{3}\right)^{-2} = 1/(e^{3})^{2} = 1/e^6\)

Solving Part (d)

The exponent of zero for any non zero number is one. Thus, e^0 = 1

Key concepts

Simplifying Expressions

When faced with complex mathematical expressions involving exponents, the key is to simplify them in a step-by-step manner. Simplifying expressions means reducing them to their simplest form without changing their value. This process often involves combining like terms and applying exponent rules. For instance, when you have the same base, such as the natural number 'e', raised to different powers and multiplied together, you simplify the expression by adding the exponents. With practice and understanding of the fundamental exponent rules, simplifying expressions becomes a more intuitive process.

Considering the textbook exercise, we see how this simplification is done with \(e^3\) and \(e^4\). The base 'e' remains the same while the exponents 3 and 4 are added together, resulting in \(e^7\). By following these guidelines for simplification, students can methodically approach and easily solve similar problems.

Exponent Rules

Exponent rules are like the roadmap for navigating problems involving powers. There are several key rules to remember. The Product of Powers rule indicates that when multiplying two powers with the same base, you keep the base and add the exponents. As seen in the example \(e^3 \cdot e^4\), this results in \(e^{3+4}\) or \(e^7\). Conversely, the Power of a Power rule, used in \(e^3\)^4, tells us to multiply the exponents, giving \(e^{3*4}\) or \(e^{12}\).

Understanding these rules provides a foundation to tackle more complex expressions and to ensure that the simplification is performed correctly. Always refer to these basic rules when simplifying expressions with exponents to avoid errors in calculation.

Negative Exponents

Negative exponents might initially be counterintuitive, but they play a critical role in understanding the reciprocal relationship in mathematics. The negative exponent rule states that a number raised to a negative exponent is equivalent to one over that number raised to the corresponding positive exponent. For example, \(e^3\)^{-2} changes to \(1/e^6\) when simplified. This indicates that rather than the number growing larger, due to the negative exponent, it represents a fraction of unity.

The advantage of mastering the negative exponent concept is that it enriches your mathematical toolkit, allowing you to confidently transform and simplify equations involving division and inverse relationships. Remember: negative exponents do not make the number negative; they place it under one in a fraction form.

Zero Exponent Property

The zero exponent property is a simple yet vital exponent rule which states that any non-zero number raised to the power of zero equals one. It’s an often surprising rule for many learners, but it simplifies the work greatly. This universal property is demonstrated in the exercise, where \(e^0\) simplifies directly to 1. It does not matter how large or small the base is; if its exponent is zero, the value is always one.

Embracing this property can drastically simplify expressions and allows you to focus on other parts of an equation where the manipulation might be more complex. The zero exponent property serves as an excellent reminder that sometimes the most powerful changes in mathematics come from the simplest rules.