Problem 1 - A Graphical Approach to College Algebra

Question

Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{10 x^{6}}{5 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(2x^3\).

Step 1: Apply the Division of Coefficients

The expression we need to simplify is \( \frac{10x^6}{5x^3} \). Start by dividing the coefficients, which are the numbers in front of the variables. Here, divide 10 by 5, resulting in 2.

Step 2: Simplify the Variable Part by Applying the Laws of Exponents

Now, focus on the variables. Recall the property of exponents \( \frac{x^a}{x^b} = x^{a-b} \). Applying this to \( \frac{x^6}{x^3} \) gives \( x^{6-3} = x^3 \).

Step 3: Combine the Simplified Parts

Combine the results from Steps 1 and 2. The coefficient simplifies to 2, and the variable simplifies to \( x^3 \). Put these together to get \( 2x^3 \).

Key Concepts

Laws of Exponents

Exponents are a useful mathematical tool for dealing with expressions that involve repeated multiplication. When simplifying algebraic expressions, often you will need to manipulate powers. The primary laws of exponents help us in this process:

• Product Rule: When multiplying terms with the same base, add the exponents: \( x^a \times x^b = x^{a+b} \).
• Quotient Rule: When dividing terms with the same base, subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).
• Power of a Power: When raising an exponent to another power, multiply the exponents: \( (x^a)^b = x^{a\cdot b} \).

In the exercise, the quotient rule is applied to simplify \( \frac{x^6}{x^3} \). By subtracting the exponents, 6 and 3, we find \( x^{6-3} \), which simplifies to \( x^3 \). This process makes it easier to handle complex algebraic expressions by breaking them down into simpler parts.

Division of Coefficients

Coefficients are the numerical parts of terms in algebraic expressions. Simplifying expressions sometimes involves dividing these coefficients. In our exercise, we started with \( \frac{10x^6}{5x^3} \). Here, the coefficients are 10 and 5.

Divide the coefficient of the numerator by the coefficient of the denominator: \( \frac{10}{5} = 2 \). This represents the part of the expression that is independent of the variable. When you have two numbers and wish to simplify them, treat them as though you're dividing regular numbers, which keeps calculations straightforward.

It’s crucial to simplify coefficients first as it helps streamline the rest of the expression, making the next operations with variables much simpler. This step is an essential part of simplifying expressions neatly and efficiently.

Simplification of Expressions

Simplification of expressions involves reducing them to their simplest form while maintaining their value. This often includes simplifying both coefficients and variables. Let’s apply this to the given expression.

Start by simplifying the coefficients: \( \frac{10}{5} = 2 \). This gives us the simplified numerical factor.

• Then, apply the laws of exponents to the variables: \( \frac{x^6}{x^3} = x^{6-3} = x^3 \).
• Now combine the simplified parts: the coefficient 2 and the power \( x^3 \), resulting in \( 2x^3 \).

By systematically working through each component of an algebraic expression, we can achieve the simplest form. Simplifying expressions not only makes them easier to work with but also enhances understanding of the relationship and proportion of their elements. It leads to more efficient problem-solving and clearer insights across various algebraic applications.