Problem 43 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{x^{2}}{18} \cdot \frac{3}{x^{3}} $$

Step-by-Step Solution

Answer

Based on the given step-by-step solution, the expression \(\frac{x^2}{18} \cdot \frac{3}{x^3}\) simplifies to \(\frac{1}{6x}\). To do this, we multiplied the numerators, multiplied the denominators, and then canceled out any common factors between them.

Step 1: Multiply the numerators

To multiply the numerators, multiply \(x^2\) and \(3\) together: $$ (x^2)(3) = 3x^2 $$

Step 2: Multiply the denominators

To multiply the denominators, multiply \(18\) and \(x^3\) together: $$ (18)(x^3) = 18x^3 $$

Step 3: Simplify the expression

Now, form the fraction with the multiplied numerators and denominators: $$ \frac{3x^2}{18x^3} $$ To simplify, cancel out the common factors between the numerator and the denominator. In this case, there's a common factor of \(3\) and \(x^2\): $$ \frac{3x^2}{18x^3} = \frac{3}{3} \cdot \frac{1}{6} \cdot \frac{x^2}{x^2} \cdot \frac{1}{x} $$ After canceling out the common factors, the simplified expression is: $$ \frac{1}{6x} $$

Key Concepts

Algebraic Fractions

In algebra, working with fractions that contain variables is quite common. These are called algebraic fractions. The structure of algebraic fractions is similar to numerical fractions, except that they can include variables in the numerator, denominator, or both. Understanding algebraic fractions is crucial in algebra because they often show up in equations and must be handled with care.

To handle algebraic fractions effectively, keep in mind that:

• The operations you use with numerical fractions (addition, subtraction, multiplication, division) work similarly with algebraic fractions.
• Variables behave like numbers, so you can apply the same rules for manipulation, like distributing, factoring, and canceling common factors.
• In some cases, like multiplication and division, it might be necessary to simplify the fractions to make operations easier.

By mastering these steps with algebraic fractions, you'll be able to solve more complex algebraic problems with confidence.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. When you simplify, you are essentially making it easier to understand or compare the fraction by reducing its numerator and denominator to their smallest possible values.

When simplifying:

• Look for common factors in the numerator and the denominator. These are numbers (or variables) that divide evenly into both.
• Cancel out these common factors. For example, in the fraction \(\frac{3x^2}{18x^3}\), both the numerator and the denominator have a factor of 3. Cancelling this common factor simplifies the fraction.
• Examine variable factors. In our example, \(x^2\) is a common factor in the numerator and the part of the denominator, so it can be canceled as well.

After removing all possible common factors, the fraction is in its simplest form, such as \(\frac{1}{6x}\) in the example. This helps not only in making calculations easier but also in showing the solution more clearly.

Exponent Rules

Exponent rules are critical when dealing with algebraic expressions, especially those containing powers of variables. Understanding these rules allows us to simplify expressions and manage operations involving exponents efficiently.

Some basic rules to remember:

• Product of Powers: When multiplying terms with the same base, add their exponents, e.g., \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing terms with the same base, subtract the exponents, e.g., \(\frac{a^m}{a^n}= a^{m-n}\).
• Negative Exponent: A negative exponent means taking the reciprocal of the base raised to the positive exponent, e.g., \(a^{-n} = \frac{1}{a^n}\).

In the exercise, we used these rules to simplify \(\frac{x^2}{x^3}\) by applying the quotient of powers rule, resulting in \(x^{2-3} = x^{-1}\). This allowed us to rewrite the expression as \(\frac{1}{x}\), critical for reaching the solution \(\frac{1}{6x}\). Knowing how to apply these rules not only simplifies complex algebraic expressions but also helps in solving algebraic problems more efficiently.