Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\frac{x^{14}}{x^{-7}}\) is \(x^{21}\).

Step 1: Identify the base and the exponents

In the expression \(\frac{x^{14}}{x^{-7}}\), 'x' is the base and '14' is the exponent of the numerator and '-7' is the exponent of the denominator.

Step 2: Use rule of exponents for division

The law of exponents says that when you divide like bases, you subtract the exponent of the denominator from the exponent of the numerator. So, A^m / A^n = A^(m-n). Using this rule, the expression becomes: \(x^{14-(-7)}\).

Step 3: Simplify the exponent

Subtract -7 from 14 to simplify the exponent. The expression then becomes: \(x^{14+7}\).

Step 4: Final Simplification

Now it's just a basic arithmetic to add the exponents. So the final simplified expression is: \(x^{21}\).

Key Concepts

Understanding the Law of Exponents

The law of exponents is a fundamental concept in algebra that provides rules for manipulating exponential expressions. It simplifies the process of working with powers by offering a structured way to combine them. There are several rules, but one of the most important is the division rule. When dividing exponential expressions with the same base, we can subtract the exponents according to the formula:

• If we have \(A^m \div A^n\), we rewrite it as \(A^{m-n}\).
• This rule helps to reduce complex expressions into simpler forms.

It's critical to ensure that the base is the same on both expressions you are dividing. If the bases differ, this rule of exponents cannot be applied directly.

Simplification of Exponential Expressions

Simplification in math means making an expression easier to understand or use. For exponential expressions, simplification involves reducing the number of terms and operations by using the laws of exponents.

• In the given expression \(\frac{x^{14}}{x^{-7}}\), we see that both the numerator and the denominator have the same base \(x\).
• The exponents can be simplified by applying the relevant rules, leading to a cleaner and more manageable form.

The main aim is to perform operations such as addition or subtraction of exponents where applicable, and to rewrite the expression in its most basic form. This makes it easier for students and professionals to work with these expressions in more complex equations.

Mastering Division of Exponents

The division of exponents is a direct application of the law of exponents. It's particularly useful when managing fractions of powers. In our example, we have \(\frac{x^{14}}{x^{-7}}\).
The steps to solve it involve:

• Identifying the base \(x\) and the exponents 14 and -7.
• Using the division rule: subtract the exponent in the denominator from the exponent in the numerator.
• This transforms our expression to \(x^{14-(-7)}\), which further simplifies to \(x^{14+7}\).
• Finally, calculate \(14 + 7\) to obtain the simplified exponent of 21.

Thus, the original problem simplifies to \(x^{21}\). Understanding this process is crucial as it lays the groundwork for dealing with more sophisticated algebraic expressions efficiently.