Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\frac{x^{-2} y}{x y^{2}}\)

Short answer

\[\frac{1}{x^{3}y}\]

Step-by-step solution

Simplify the expression by separating the variables

First, rewrite the expression by separating the variables with negative and positive exponents:\[\frac{x^{-2} y}{x y^{2}} = \frac{x^{-2}}{x} \times \frac{y}{y^{2}}\]

Simplify the individual fractions

Next, simplify each fraction by subtracting exponents for the same bases:\[\frac{x^{-2}}{x} = x^{-2-1} = x^{-3}\] and \[\frac{y}{y^{2}} = y^{1-2} = y^{-1}\]

Express with positive exponents

Rewrite the expression so that all exponents are positive:\[x^{-3} = \frac{1}{x^{3}}\] and \[y^{-1} = \frac{1}{y}\]

Combine the simplified fractions

Combine the fractions to get one simplified expression:\[\frac{1}{x^{3}} \times \frac{1}{y} = \frac{1}{x^{3}y}\]

Key concepts

Positive Exponents

Exponents are used to show how many times a number, called the base, is multiplied by itself. When you have a positive exponent, it means you're multiplying the base by itself a certain number of times. For example, in the expression \(a^n\), where \(a\) is the base and \(n\) is the exponent, \(n\) is positive. This means you multiply \(a\) by itself \(n\) times.
For instance:
\(2^3 = 2 \times 2 \times 2 = 8\)

When you're working with algebraic expressions, you may see variables as bases. It's essential to remember that the rules for positive exponents are the same for numbers and variables. If you have \(x^2\), it means \(x\) multiplied by itself, which is \(x \times x\).
Understanding positive exponents helps us simplify expressions effectively, making complex operations more manageable.

Negative Exponents

When you see a negative exponent, it means you need to take the reciprocal of the base and then apply the positive exponent. The reciprocal of a number is simply \(1\) divided by that number. For example, \(a^{-n} = \frac{1}{a^n}\).
This might sound complicated, but it’s not. Here’s an example to understand this better:
\(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)

In the context of variables, if you have \(x^{-2}\), it means \(\frac{1}{x^2}\). We use this rule to make all exponents positive, which often makes the expressions easier to handle. This becomes essential in algebraic expressions when simplifying them. Remember, converting negative exponents to positive ones and then simplifying makes the entire calculation more straightforward.

Fraction Simplification

Simplifying fractions is a key skill in algebra. The goal is to make the fraction as simple as possible, which means reducing it to its lowest terms. To do this, you divide the numerator and the denominator by their greatest common divisor (GCD).
When dealing with algebraic fractions, you often need to simplify both the coefficients (numbers) and the variables. For example:
\(\frac{2x^3}{4x^2} = \frac{2}{4} \times \frac{x^3}{x^2}\) simplifies to \(\frac{1}{2} \times x^{3-2} = \frac{1}{2} \times x = \frac{x}{2}\)

Notice that we subtract the exponents of the same base and divide the numerical coefficients by their GCD. Simplifying fractions ensures that the expression is as manageable as possible, making further operations easier.

Algebraic Expressions

An algebraic expression includes numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions are the building blocks of algebra. Understanding how to manipulate them is crucial for solving algebraic problems.

For example, consider the expression \(\frac{x^{-2} y}{x y^{2}}\). To simplify it, we must:
1. Separate the variables: \(\frac{x^{-2}}{x} \times \frac{y}{y^2}\)
2. Simplify each fraction: \(x^{-2-1} = x^{-3}\) and \(y^{1-2} = y^{-1}\)
3. Convert negative exponents to positive: \(x^{-3} = \frac{1}{x^3}\) and \(y^{-1} = \frac{1}{y}\)
4. Combine the simplified fractions: \(\frac{1}{x^3} \times \frac{1}{y} = \frac{1}{x^3 y}\)
Mastering these steps allows students to handle more complex algebraic expressions with confidence.