Problem 81 - Algebra for College Students

Question

Express each of the following as a single fraction involving positive exponents only. \(x^{-1} y^{-2}-x y^{-1}\)

Step-by-Step Solution

Answer

The expression is simplified to \(\frac{1 - xy}{xy^{2}}\).

Step 1: Express Negative Exponents as Positive

Firstly, recall that negative exponents represent reciprocals, e.g., \(a^{-n} = \frac{1}{a^{n}}\). Apply this concept to each term in the expression to rewrite it with positive exponents. The expression \(x^{-1} y^{-2}\) can be rewritten as \(\frac{1}{xy^{2}}\) and \(-xy^{-1}\) becomes \(-\frac{x}{y}\).

Step 2: Find Common Denominator

Identify a common denominator for the fractions \(\frac{1}{xy^{2}}\) and \(-\frac{x}{y}\). The least common denominator (LCD) is \(xy^{2}\).

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite each term with the common denominator \(xy^{2}\). The first term is already over the common denominator: \(\frac{1}{xy^{2}}\). To convert the second term \(-\frac{x}{y}\) to have the denominator \(xy^{2}\), multiply both the numerator and the denominator by \(y\): \(-\frac{x}{y} \times \frac{y}{y} = -\frac{xy}{y^{2}}\).

Step 4: Combine the Fractions

Combine \(\frac{1}{xy^{2}}\) and \(-\frac{xy}{y^{2}}\) over the common denominator: \(\frac{1 - xy}{xy^{2}}\).

Step 5: Simplify the Expression

Examine the numerator and denominator to see if there are common factors that can be simplified. There are no common factors in the numerator and the denominator, so the final simplified expression remains: \(\frac{1 - xy}{xy^{2}}\).

Key Concepts

Negative Exponents

Negative exponents can sometimes be confusing, but they're straightforward once you understand the rule. When you see a negative exponent, it simply means you'll be dealing with a reciprocal. For example, if you have something like \(a^{-n}\), this is equivalent to \(\frac{1}{a^n}\). The negative sign indicates that the base \(a\) is moved to the opposite side of the fraction.

• \(x^{-1} = \frac{1}{x}\): moves the base \(x\) to the denominator.
• \(y^{-2} = \frac{1}{y^2}\): means \(y^2\) is in the denominator as a square.

So, when you apply this rule to an expression like \(x^{-1}y^{-2}\), it becomes \(\frac{1}{xy^2}\). Remember, turning negative exponents into positive ones simplifies the work of handling algebraic expressions.

Common Denominator

Finding a common denominator is a fundamental step when dealing with fractions. It helps in adding or subtracting fractions by giving them a shared basis. If you have fractions with different denominators, you can't simply add or subtract them.To find a common denominator, you look for the least common multiple (LCM) of the denominators involved. In the expression we’re working with, we have the fractions \(\frac{1}{xy^2}\) and \(-\frac{x}{y}\). The LCM of \(xy^2\) and \(y\) is \(xy^2\). This means both fractions should be rewritten to have \(xy^2\) as their denominator before combining them. Use this approach anytime you process similar problems involving fractions.

Simplifying Expressions

Simplifying expressions requires careful inspection to combine and reduce terms in the simplest form. For the fractions aligned with a common denominator, like \(\frac{1}{xy^2}\) and \(-\frac{xy}{y^2}\), combine them under this shared denominator. Simplification often involves:

• Combining numerators over a shared denominator.
• Checking for and eliminating any common factors.

After rewriting and combining, you have \(\frac{1 - xy}{xy^2}\). Here, the expression is already simplified because there are no shared factors between \(1 - xy\) and \(xy^2\). Whenever simplifying, always verify if further reduction is necessary by checking for common factors.

Positive Exponents

Positive exponents are as straightforward as they sound. In contrast to negative exponents, they imply straightforward multiplication rather than reciprocals. For instance, \(x^1 = x\), and more generally, \(x^n\) means multiplying \(x\) by itself \(n\) times.In the context of our expression, converting negative exponents to positive ones makes the expression easier to interpret and work with. For instance, changing \(x^{-1}\) and \(y^{-2}\) to \(\frac{1}{x}\) and \(\frac{1}{y^2}\) involves ensuring the power is positive, thus simplifying subsequent calculations. Hence, working with positive exponents allows you to follow standard algebraic manipulation rules seamlessly.