Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{-48 x^{3}}{-12 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4x \).

Step 1: Divide the numbers

Divide -48 by -12 to get 4.

Step 2: Subtract powers of \( x \)

According to the rule \( x^{m} / x^{n} = x^{m-n}\), subtract 2 from 3 to get 1. The subtraction here is due to the division in the initial problem.

Step 3: Write the simplified expression

Combine the results from steps 1 and 2 to form the simplified expression, which is \( 4x^{1} \) or simply \( 4x \).

Key Concepts

Rational Expressions

Rational expressions are a fundamental concept in algebra that involves fractions formed by dividing two polynomials. They tend to look tricky at first, but by treating them like fractions, we can make them much simpler. In the given problem, the expression \( \frac{-48 x^{3}}{-12 x^{2}} \) is a rational expression because it involves division where both the numerator and denominator contain polynomials. To simplify rational expressions, you can usually:

• Divide the coefficients (numerical values) in the numerator and denominator.
• Apply the rules of exponents for the variables involved.

Rational expressions are easier to manage when broken into simpler components. Always aim to make the expression as simple as possible by reducing both numbers and variables.

Exponents

Exponents are a powerful tool in algebra for representing repeated multiplication. When you see a term like \( x^{m} \), it means \( x \) is multiplied by itself \( m \) times. In the context of our expression \( \frac{-48 x^{3}}{-12 x^{2}} \), the exponents show us how many times each \( x \) appears in both the numerator and the denominator.To simplify exponents during division, we use the property:

• \( x^{m} / x^{n} = x^{m-n} \)

This is because dividing by \( x \) is essentially the same as cancelling out one \( x \) at a time from the top and the bottom. So, when we have \( x^{3}/x^{2} \), we subtract the exponent in the denominator from the exponent in the numerator to get \( x^{1} \), equivalent to \( x \). This operation reduces the complexity of expressions with exponents and makes it easier to interpret the algebraic solution.

Division of Like Bases

When dividing terms with the same base, such as \( x \, \text{in} \, x^{m}/x^{n} \), it's important to understand the rules of subtraction applied to the exponents. The term "like bases" refers to variables that are identical, thus making them suitable for simplification rules.In our example, with bases that are both \( x \), we can simplify considerably by:

• Dividing the coefficients: \( \frac{-48}{-12} = 4 \).
• Subtracting the exponents: \( 3 - 2 = 1 \), so \( x^{3}/x^{2} = x^{1} \).

This process highlights the elegance of mathematical rules in algebra. By reducing both the numbers (coefficients) and powers (exponents) independently, we arrive at a cleaner and more straightforward expression. Mastering division of like bases involves recognizing common variables and applying exponent subtraction effectively to simplify expressions.