Problem 61 - Beginning and Intermediate Algebra

Question

Simplify. Assume that the variables represent nonzero integers. $$\frac{x^{7 b}}{x^{4 b}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{3b}\).

Step 1: Identify the Division Rule for Exponents

When dividing exponential expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator. In this exercise, the base is x, and the exponents are 7b and 4b. So, the rule we will apply is: \(x^{a}/x^{b} = x^{a-b}\)

Step 2: Apply the Division Rule for Exponents

Applying the division rule for exponents to the given expression, we get: \(\dfrac{x^{7b}}{x^{4b}} = x^{(7b - 4b)}\)

Step 3: Simplify the Exponents

Subtract the exponents: \(x^{(7b - 4b)} = x^{3b}\)

Step 4: Write the Final Answer

The simplified expression is: \(x^{3b}\)

Key Concepts

Simplifying Expressions

Sometimes when working with algebraic expressions, you'll find expressions with exponents that you need to simplify. Simplifying expressions involves breaking them down to their simplest form, which often makes solving problems easier. One common method of simplifying expressions is to use exponent rules. These rules help us to combine like terms, especially when dealing with expressions with the same base. Simplifying makes algebra problems less complex, enabling you to see results and solutions more clearly. Remember: always ensure your expressions are correct by following the proper rules, like the division rule, which we'll discuss next.

Division Rule

The division rule is an essential tool when simplifying expressions involving exponents. Specifically, it deals with dividing exponents that have the same base. According to this rule, when you have a fraction where the numerator and the denominator have the same base, you can simplify it by subtracting the exponent in the denominator from the exponent in the numerator. For example, if you have an expression like \(\frac{x^m}{x^n}\), it can be simplified to \(x^{m-n}\). This is because the base, \(x\), doesn't change; only the exponents are affected.

• Key point: Base stays the same.
• Action: Subtract the exponent in the denominator from the numerator.

This rule helps make expressions cleaner and simpler, allowing you to understand and work with them more efficiently.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators. These can include terms like addition, subtraction, multiplication, division, and exponents. They form the basis of many algebra problems.Understanding algebraic expressions involves recognizing these different components and how they interact. Variables, like \(x\), can represent numbers, and exponents can indicate how many times a variable is multiplied by itself.

• Terms are separated by addition or subtraction.
• Similar terms can often be combined for simplification.

By mastering the various rules applied to algebraic expressions, such as those for exponents (like the division rule), you can simplify and solve equations with confidence. Practicing these rules aids in a deeper understanding of how mathematical operations connect and work together.