Problem 53 - College Algebra

Question

Simplify each exponential expression $$ \frac{14 b^{7}}{7 b^{14}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(2/b^{7}\).

Step 1: Simplify the coefficient

In the given expression \(\frac{14b^{7}}{7b^{14}}\), first simplify the coefficient i.e., \(\frac{14}{7} = 2\), yielding \(2b^{7}/b^{14}\).

Step 2: Apply the quotient of powers rule

Next, apply the quotient of powers rule, which says you can subtract the exponents when dividing like bases. So, subtract the exponent of the base \(b\) in the denominator from that in the numerator: \(7 - 14 = -7\), yielding \(2b^{ - 7}\).

Step 3: Present in positive exponent

The negative exponent in the base \(b\) can be rewritten as \(1/b^{7}\) to make the exponent positive, yielding \(2/b^{7}\).

Key Concepts

Quotient of Powers Rule

One of the foundational components of working with exponential expressions is understanding the quotient of powers rule. This rule is essential when simplifying expressions that involve dividing powers with the same base. When you divide exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
To illustrate, consider an expression \(\frac{a^m}{a^n}\), where \(a\) is the base and \(m\) and \(n\) are the exponents. According to the quotient of powers rule, this expression can be simplified to \(a^{m-n}\).
For example, with the problem \(\frac{14 b^{7}}{7 b^{14}}\), once we simplify the coefficients separately, we have \(\frac{2 b^{7}}{b^{14}}\). Here, \(b\) is the common base, and we can apply the quotient of powers rule: \(7-14 = -7\). Hence, the expression simplifies to \(2b^{-7}\), reducing the initial equation to a simpler form using this powerful rule.

Negative Exponents

The concept of negative exponents plays a vital role in performing algebraic simplification. \(a^{-n}\) is equivalent to \(1/a^n\), meaning that a term with a negative exponent represents the reciprocal of that term with a positive exponent. Such a representation can dramatically change the outlook of an expression and are useful in various algebraic manipulations.
Consider the simplified expression \(2b^{-7}\) derived from applying the quotient of powers rule. This expression contains \(b^{-7}\), and to express it with a positive exponent, we rewrite it as \(2 \times (1/b^7)\). Understanding the nature of negative exponents not only aids in simplifying expressions but ensures they are correctly interpreted in the context of the problem.

Algebraic Simplification

Algebraic simplification encompasses various rules and operations that enable us to condense and simplify algebraic expressions. This process usually includes steps like combining like terms, applying exponent rules (like the quotient of powers rule), and transforming negative exponents into positive ones, as seen in our exercise.
In the case of our example \(\frac{14 b^{7}}{7 b^{14}}\), each of these steps is sequentially applied. First, we simplify the coefficients which are the numerical parts of the terms, reducing \(\frac{14}{7}\) to 2. Then, we use the quotient of powers rule to manage the variables with exponents, and we finally reinterpret negative exponents to present the terms in a more conventional positive exponent format. The resultant expression, \(2/b^{7}\), is much more straightforward and thus demonstrates the effectiveness of algebraic simplification.