Problem 160 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. In parts (a) and (b), complete each statement. a. \(\frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{?}\) b. \(\frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?

Step-by-Step Solution

Answer

For question (a), the missing exponent after simplification is 4, and for question (b), it is 6. In general, the rule of exponents when dividing expressions with the same base is to subtract the exponent of the denominator from the exponent of the numerator.

Step 1: Solving part (a)

The problem in part (a) is \(\frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{?}\). When simplifying, subtract the exponent of \(b\) in the denominator from the exponent of \(b\) in the numerator. This gives the result as \(b^{(7-3)} = b^{4}\).

Step 2: Solving part (b)

The problem in part (b) is \(\frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?}\). Following the same steps as in part (a), subtract the exponent of \(b\) in the denominator from the exponent of \(b\) in the numerator. This gives the result as \(b^{(8-2)} = b^{6}\).

Step 3: Answering part (c)

Based on the above steps, a generalization can be made: When dividing exponential expressions with the same base, you should subtract the exponent in the denominator from the exponent in the numerator.

Key Concepts

Simplifying Exponents

Understanding how to simplify exponents is essential for mastering algebra. When we have an expression like \(\frac{b^{7}}{b^{3}}\), we are dealing with a division of two powers with the same base, which is 'b' in this case. Simplifying such expressions involves using the laws of exponents, particularly the rule for division: subtract the exponent in the denominator from the exponent in the numerator.

So, for \(\frac{b^{7}}{b^{3}}\), we subtract 3 from 7, giving us \(b^{4}\). Simplification is all about reducing expressions to their most basic form, which not only makes them easier to work with but also helps in understanding the underlying relationships between numbers and variables in algebra.

Laws of Exponents

The laws of exponents are a set of rules that describe how to handle operations involving powers. These include multiplying powers with the same base, dividing powers with the same base, raising a power to a power, and more. When we divide powers with the same base, like \(\frac{b^{8}}{b^{2}}\), we use one of these laws, resulting in \(b^{6}\) after subtracting the exponents.

Key Laws of Exponents

• Product of powers: \(b^{m} \cdot b^{n} = b^{m+n}\)
• Quotient of powers: \(\frac{b^{m}}{b^{n}} = b^{m-n}\), when \(m > n\)
• Power of a power: \((b^{m})^{n} = b^{mn}\)
• Power of a product: \((ab)^{n} = a^{n}b^{n}\)
• Power of a quotient: \(\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}\)

By applying these laws properly, you can simplify complex exponential expressions to solve various algebraic problems.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and involve variables, numbers, and operations. The ability to manipulate these expressions according to algebraic principles allows for the solving of equations and simplification of complex formulas. Simplifying exponential expressions, as seen in the example exercise, is a practical application of working with algebraic expressions.

An algebraic expression can represent a wide range of relationships and is often used to describe real-world scenarios mathematically. Mastery of algebraic expressions enables students to progress to more advanced topics in mathematics and apply these concepts to solve real-world problems.

College Algebra

College algebra is a course that often serves as a gateway to higher-level mathematics courses. It builds on the foundation of high school algebra and delves deeper into the study of functions, complex numbers, and various algebraic techniques. A solid understanding of exponential expressions and the laws of exponents is a fundamental part of college algebra.

The exercise example illustrates a type of problem that students might encounter in a college algebra course. It is not just about getting to the solution, but about understanding the process and being able to generalize and apply the concepts to a variety of problems. Recognizing patterns, such as subtracting exponents when dividing like bases, is a critical skill that will be revisited in many future algebraic concepts.