Problem 47 - Algebra 2

Question

Simplify each expression. $$ \frac{x y}{\sqrt{z}} $$

Step-by-Step Solution

Answer

\( \frac{xy \sqrt{z}}{z} \)

Step 1: Identify the components

The given expression is \( \frac{x y}{\sqrt{z}} \). It consists of a numerator \( xy \) and a denominator \( \sqrt{z} \).

Step 2: Rationalize the denominator

To simplify expressions with square roots in the denominator, you rationalize the denominator by multiplying both the numerator and the denominator by the square root present in the denominator. Here, multiply both by \( \sqrt{z} \): \[ \frac{xy}{\sqrt{z}} \times \frac{\sqrt{z}}{\sqrt{z}} = \frac{xy \sqrt{z}}{z} \].

Step 3: Simplify the expression

After rationalizing, the expression becomes \( \frac{xy \sqrt{z}}{z} \). Since there are no further simplifications possible without additional information about \( x \), \( y \), or \( z \), this is the simplified form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making them easier to work with or understand. It's like tidying your room - you want it to be clean and organized. When you're given an expression, the goal is to reduce it to its most basic form without changing its value. In this exercise, we simplified the expression by dealing with the square root in the denominator, which we'll discuss in the next section. Here are some general steps to keep in mind when simplifying expressions:

• Identify like terms and combine them. Like terms have the same variable part with the same exponent.
• Look for opportunities to factor expressions. Factoring can often lead to cancellation of terms.
• Rationalize denominators when necessary, especially when a square root appears in the denominator.
• Always check for any common factors between the numerator and the denominator that can be simplified.

Following these steps will help in simplifying all sorts of algebraic expressions you encounter.

Square Roots

Square roots are mathematical operations that help find a number which when multiplied by itself gives the original number. In simple terms, the square root of a number is a value that, when squared, gives the original number back. For example, the square root of 9 is 3 because 3 times 3 equals 9. In algebra, square roots can often appear in fractions, such as in our original exercise. When they do, it is common practice to rationalize the expression by clearing the square root from the denominator. This makes further calculations easier and the expressions cleaner. Rationalizing the denominator involves:

• Identifying the square root in the denominator.
• Multiplying both the numerator and the denominator by the same square root to eliminate it from the denominator.

This technique ensures that we are left with a rational number in the denominator, making the expression simpler and easier to understand.

Algebraic Fractions

Algebraic fractions feature variables in the numerator, denominator, or both. They function similarly to regular fractions, but with an extra layer of complexity due to the presence of variables. In algebraic fractions, all the usual fraction rules apply. You can add, subtract, multiply, and divide them, but you must respect the rules of algebra throughout. When simplifying algebraic fractions, consider:

• Identifying common factors in both the numerator and the denominator and canceling them out when possible.
• Ensuring any square roots or irrational numbers are managed appropriately, often through rationalization.
• Recognizing that an expression is fully simplified when it cannot be reduced any further without additional information about the variables involved.

Mastering algebraic fractions is key to tackling many algebraic equations and expressions efficiently.