Problem 160 - Algebra and Trigonometry

Question

In parts (a) and (b), complete each statement. $$ \text { a. } \frac{b^{7}}{b^{3}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}=b^{?} $$ $$ \text { b. } \frac{b^{8}}{b^{2}}=\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}=b^{?} $$ 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

Expression (a) is equivalent to \( b^{4} \) and expression (b) is equivalent to \( b^{6} \). The general rule for division of exponential expressions with the same base is to subtract the exponent on the denominator from the exponent on the numerator.

Step 1: Simplify Expression (a)

The expression is \( \frac{b^{7}}{b^{3}} \). To simplify it, subtract the exponent in the denominator from the exponent in the numerator. This is done as follows: \( b^{(7-3)} \). Hence, the expression would be equivalent to \( b^{4} \)

Step 2: Simplify Expression (b)

The expression is \( \frac{b^{8}}{b^{2}} \). Similar to step 1, subtract the exponent in the denominator from the one in the numerator. This results in: \( b^{(8-2)} \). Therefore, the expression is equivalent to \( b^{6} \)

Step 3: Generalize the Rule

Having simplified both expressions, it can be said that the general rule for dividing exponential expressions with the same base is to subtract the exponent of the divisor (denominator) from the exponent of the dividend (numerator). The base remains the same while its power is calculated as: \( base^{(exponent_{dividend} - exponent_{divisor})} \)

Key Concepts

Understanding Exponentiation

Exponentiation refers to the process of raising a number, known as the base, to the power of an exponent. In simple terms, it's expressing a number to be multiplied by itself a certain number of times. For example, in the expression \( b^7 \), \( b \) is the base, and 7 is the exponent, indicating that \( b \) is multiplied by itself 7 times:

• \( b^7 = b \times b \times b \times b \times b \times b \times b \)

This fundamental concept is important for understanding how numbers behave when scaled up exponentially. Exponents allow us to write large numbers in a compact form, facilitating easier computation and comprehension.
In mathematical analysis, exponentiation is a cornerstone that applies to various branches such as algebra and calculus.

Division of Exponents

When dividing two exponential expressions that share the same base, a special rule comes into play: the quotient of powers rule. This rule states that you subtract the exponent in the denominator from the exponent in the numerator while keeping the base unchanged. Here's how it looks: Let's analyze the provided example:

• \( \frac{b^7}{b^3} = b^{7-3} = b^4 \)
• \( \frac{b^8}{b^2} = b^{8-2} = b^6 \)

It's important to remember that this division rule only works when the bases are identical. Thus, it simplifies calculations and is a fundamental technique in algebra when solving equations and expressions.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form without changing their value. With exponential expressions, this often means applying the rules of exponents. In the context of exponentiation:

• The expression \( \frac{b^7}{b^3} \) is simplified by using the exponent rules: subtract the exponents to get \( b^4 \).
• This means the simplified expression retains the same meaning but becomes much easier to work with and understand.

Simplification helps in making algebraic expressions more manageable, easing further mathematical operations or transformations.

Core Mathematical Rules

Core mathematical rules frame how we manipulate numbers in algebraic processes, including how we handle operations on exponents. These rules ensure consistency and accuracy in calculations. The key rule used in dividing exponents with the same base is the quotient of powers rule:

• Subtract the exponent of the denominator from the exponent of the numerator: \( b^{m-n} \)

Such rules are not just theoretical; they are enforced because they make sense procedurally and logically, ensuring that expressions evolve correctly through mathematical operations. Understanding and applying these rules is vital for solving more advanced problems in mathematics, particularly as complexity increases. By mastering these rules in exponentiation and division, students can effectively break down and solve exponential equations.