Problem 92 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{5 x^{9}}{x^{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 5x^6 \).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{5x^9}{x^3} \). Simplifying involves reducing the expression to its simplest form.

Step 2: Apply the Division Rule for Exponents

Recall that when you divide terms with the same base, you subtract the exponents: \( x^a \div x^b = x^{a-b} \). Here, the base is \( x \) and the exponents are 9 and 3.

Step 3: Perform the Subtraction of Exponents

Subtract the exponents of \( x \): \( x^{9-3} = x^6 \). This simplifies our expression to just \( x^6 \).

Step 4: Simplify the Coefficient

The coefficient \( 5 \) remains unchanged as there are no terms to combine it with. Therefore, the entire expression simplifies to \( 5x^6 \).

Key Concepts

Exponents Rules

Exponents are a fundamental concept in algebra that represent repeated multiplication of a number by itself. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the term \( x^9 \), \( x \) is the base and 9 is the exponent, meaning \( x \) is multiplied by itself 9 times.
When working with exponents, certain rules can simplify expressions significantly. These rules include:

• **Product of Powers Rule**: When multiplying like bases, add the exponents: \( x^a \cdot x^b = x^{a+b} \).
• **Power of a Power Rule**: When raising a power to another power, multiply the exponents: \( (x^a)^b = x^{a \cdot b} \).
• **Division of Powers Rule**: When dividing like bases, subtract the exponents: \( x^a \div x^b = x^{a-b} \).

Applying these rules simplifies complex expressions and is essential for solving algebraic equations efficiently.

Division of Exponents

The division of exponents is an application of one of the exponents rules that deals with simplifying expressions where a term with an exponent is divided by another term with the same base but possibly a different exponent. The rule states that when dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.
For example, consider the expression \( \frac{x^9}{x^3} \):

• The base \( x \) is the same for both terms.
• Subtract the exponents: \( 9 - 3 = 6 \).
• The simplified result is \( x^6 \).

This rule is pivotal in simplifying expressions because it systematically reduces the complexity by consolidating the terms.

Coefficients in Algebra

In algebra, coefficients are the numerical factors attached to variables. They give scale or magnitude to the variable. In the expression \( 5x^9 \), \( 5 \) is the coefficient, and it multiplies the variable part of the term. Br>

• **Constant Coefficient**: Remains unchanged unless there's an operation involving another numerical coefficient.
• **Variable Coefficient**: Multiplies the variable and takes part in operations involving algebraic expressions.

In our example, \( \frac{5x^9}{x^3} \), the number \( 5 \) is the coefficient of \( x^9 \). Since there are no other numerical coefficients to merge or adjust, it stays unchanged. Thus, in the final expression, \( 5 \) continues to serve as the coefficient for \( x^6 \). Maintaining clarity on coefficients helps in understanding the scale of expressions and ensures precise simplification.