Problem 35 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\frac{x^{14}}{x^{7}}$$

Step-by-Step Solution

Answer

\(\frac{x^{14}}{x^{7}}\) simplifies to \(x^7\).

Step 1: Identify the Base and Exponents

First, identify the base and exponent in both the numerator and denominator. Here, the base is \(x\) in both the numerator and denominator. The exponents are 14 and 7 respectively.

Step 2: Apply the Rule of Exponents

When you are dividing expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. So the new exponent is 14 - 7.

Step 3: Simplify the Expression

Perform the calculation for the new exponent. The equation simplifies to \(x^{14 - 7}\), which further simplifies to \(x^7\).

Key Concepts

Exponents

An exponent represents the number of times a base is multiplied by itself. It's written as a small number, known as the power, above and to the right of the base number. Understanding exponents is crucial, as they appear frequently in various mathematical contexts, from algebra to calculus.

For example, when you see an expression like \(x^3\), it indicates that \(x\) should be multiplied by itself three times: \(x \times x \times x\). This is more concise than writing out all the multiplications and allows for simpler manipulation of mathematical expressions, especially during operations involving multiple exponential terms.

Base and Exponent

Every exponential expression comprises two parts: the base and the exponent. The base is the number that's being multiplied by itself, and the exponent tells us how many times the multiplication occurs. In the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent.

One key characteristic of exponential expressions is that they only involve repeated multiplication of a single base. If the base changes, even if the exponent remains the same, the expressions are no longer similar and cannot be simplified in the same manner as those with like bases.

Rule of Exponents

When simplifying exponential expressions, it is essential to follow the rules of exponents. These rules help us combine or break down expressions without altering their values. One basic rule is the quotient rule, which comes into play when you divide expressions with the same base. According to the rule, you subtract the exponent in the denominator from the exponent in the numerator.

For instance, given \(a^m \div a^n\) where both the base \(a\) and the exponents \(m\) and \(n\) are the same, the expression simplifies to \(a^{m-n}\). This is powerful because it gives us a direct way to simplify what could otherwise be a very complex expression.