Problem 41 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{x^{15}}{x^{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^{12} \).

Step 1: Understand the Quotient Rule

The quotient rule for exponents states that when you divide two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). In this problem, the base is \( x \), and the exponents are 15 and 3.

Step 2: Apply the Quotient Rule

Using the quotient rule, subtract the exponent in the denominator from the exponent in the numerator: \( x^{15-3} \).

Step 3: Simplify the Expression

Subtract the exponents: \( 15 - 3 = 12 \). So, the expression simplifies to \( x^{12} \).

Key Concepts

Exponent Properties

Exponents are powerful tools in mathematics that describe how many times a number, or base, is multiplied by itself. Understanding the properties of exponents is essential for simplifying complex algebraic expressions. Here are some key properties:

• Product of Powers: When multiplying two powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).


• Quotient of Powers: This is particularly relevant to the problem at hand. It states that when dividing two powers with the same base, the exponents are subtracted: \( \frac{a^m}{a^n} = a^{m-n} \).


• Power of a Power: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

These properties allow us to manipulate and simplify expressions quickly, turning complex equations into more manageable forms.

Simplifying Expressions

Simplifying expressions involves using mathematical laws and properties, like those of exponents, to reduce expressions to their simplest form. This process helps in solving equations more efficiently and in making calculations easier.

• Identify common bases and use the exponent rules to simplify terms.


• Ensure the expression is in its simplest form—no further simplification should be possible.


• Pay careful attention to the signs of numbers and exponents, as they can affect the simplification.

By applying the quotient rule for exponents, the expression \( \frac{x^{15}}{x^{3}} \) simplifies to \( x^{12} \), demonstrating how a seemingly complex quotient can become a simple power of the base.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They can represent fundamental math principles in a general form. These expressions use letters to represent numbers and apply operations such as addition, subtraction, multiplication, and division.

• Variables: Typically letters (such as \( x \), \( y \), \( z \)) representing unknown values or changing quantities.


• Constants: Numbers that have a fixed value.


• Coefficients: Numerical factors multiplied by the variables in expressions.

Understanding how to manipulate algebraic expressions is crucial for solving equations. By using rules like the quotient rule for exponents, you can efficiently simplify expressions appearing in algebraic problems, which is essential for providing solutions to various mathematical and real-world problems.