Problem 35 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified form of the exponential expression \(\frac{x^{14}}{x^{7}}\) is \(x^7\).

Step 1: Identify the Base and Exponents

Firstly recognize the base \(x\) being raised to the power of 14 in the numerator and 7 in the denominator. The expression is \(\frac{x^{14}}{x^{7}}\)

Step 2: Apply the Rule of Exponents

Use the rule for dividing like bases \(a^n / a^m = a^{n-m}\), in this case, the base is \(x\), \(n=14\) and \(m=7\). Thus the expression becomes \(x^{14-7}\)

Step 3: Simplify the Expression

Now subtract the exponents: \(14-7=7\), thus the simplified expression is \(x^7\)

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics that helps make complex problems more manageable. When you encounter an expression like \( \frac{x^{14}}{x^{7}} \), simplifying it involves reducing the expression to a simpler form.Here, we notice that both the numerator and the denominator are powers of the same base \( x \). This hints that the expression can be simplified by reducing the exponents. To simplify expressions such as these:

• Focus on the common base across the terms.
• Perform mathematical operations that lower the complexity of exponents.

Through simplification, you transform a lengthy expression into a more streamlined form, which often represents the same value but is easier to work with, especially in further calculations or equations.

Laws of Exponents

The laws of exponents are rules that govern how exponents behave during mathematical operations. They are crucial for simplifying expressions with powers, like \( \frac{x^{14}}{x^{7}} \).One key rule is the division of like bases: when dividing terms with the same base, you subtract the exponents. Mathematically, it can be expressed as:

• \( a^n / a^m = a^{n-m} \)

In this exercise, the base \( x \) has exponents in both the numerator (14) and the denominator (7), and according to the division rule, you subtract the exponents: \( 14 - 7 \). This leads to:

• \( x^7 \)

This rule helps streamline expressions, cutting through complexities by using straightforward arithmetic on the exponents. Mastering the laws of exponents enhances problem-solving and aids in a deeper understanding of algebra.

Mathematical Operations

Mathematical operations in the context of exponents typically involve addition, subtraction, multiplication, and division. For expressions like \( \frac{x^{14}}{x^{7}} \), division plays a critical role.When you divide terms with the same base, the subtraction of exponents is the primary operation employed. Each operation is guided by specific rules which help simplify otherwise complex mathematics:

• Addition of exponents occurs when multiplying like bases: \( a^n \times a^m = a^{n+m} \)
• Subtraction of exponents, as discussed, occurs during the division: \( a^n / a^m = a^{n-m} \)
• Multiplication and division of different bases involves calculating each base separately.

Understanding these operations and rules provides clarity and proficiency in handling exponential expressions, transforming a daunting problem into an easier solution.