Problem 23 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{1}{\sqrt{12 z}}\)

Step-by-Step Solution

Answer

Rationalized expression: \( \frac{\sqrt{12z}}{12z} \).

Step 1: Identify the problem with the denominator

The expression \( \frac{1}{\sqrt{12z}} \) has a denominator with a square root, which needs to be rationalized. The goal is to eliminate the square root in the denominator.

Step 2: Multiply by a form of 1

To remove the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{12z} \). This is multiplying by a form of 1 since \( \sqrt{12z} / \sqrt{12z} = 1 \). This step will help in rationalizing the denominator.

Step 3: Multiply the expressions

Multiply \( \frac{1}{\sqrt{12z}} \) by \( \frac{\sqrt{12z}}{\sqrt{12z}} \). This gives: \[\frac{1 \cdot \sqrt{12z}}{\sqrt{12z} \cdot \sqrt{12z}} = \frac{\sqrt{12z}}{12z}.\]

Step 4: Simplify the result

The expression \( \frac{\sqrt{12z}}{12z} \) is simplified by observing that the denominator is now rational. The expression is in its simplest rational form. There's no need for further simplification since there are no common factors to cancel in the numerator and denominator.

Key Concepts

Square Roots

Square roots are important in both arithmetic and algebra, but they can sometimes make expressions look complicated. A square root, such as \( \sqrt{12z} \), is asking, "What number, when you multiply it by itself, gives you 12z?"
This branch of math often focuses on not leaving square roots in the denominator of a fraction.

• Rationalizing a denominator involves a process to eliminate squares roots or other irrational numbers from the bottom of a fraction.
• This is essential because it's easier to work with whole numbers and integers than lengthy decimals and irrational numbers.

A useful technique is to multiply both the top and bottom of the fraction by the term that will clear out the square root. This keeps the fraction equivalent to the original value.
By doing so, you can ensure that your expressions are as simplified and as clean as possible. It's like cleaning up the room before presenting it to guests—making it more presentable.

Algebraic Fractions

Algebraic fractions might look just like regular fractions, but they have variables involved, such as \( \frac{1}{\sqrt{12z}} \).
These fractions require special attention, since not only numbers but also algebraic expressions can be present both in numerators and denominators.

• One of the main goals with algebraic fractions is simplifying them to their cleanest form.
• This involves making sure the denominator does not include square roots (or other forms of complicated numbers).
• This step makes calculations easier and the expression more straightforward.

When dealing with algebraic fractions, always ensure that any manipulations respect the rules of algebra. Multiplying by a version of one, like \( \frac{\sqrt{12z}}{\sqrt{12z}} \), helps maintain the integrity of the fraction while making the denominator rational.

Simplifying Expressions

Simplifying expressions is a fundamental concept that focuses on reducing expressions to their most basic form without changing their value. Consider the expression \( \frac{\sqrt{12z}}{12z} \). After rationalizing the denominator, check for any additional simplification possibilities.

• The expression is already quite simplified if there are no common factors in the numerator and denominator.
• Ensuring the absence of square roots or complex numbers in the denominator often signals we've done the job right.
• Always look to see if additional factorization or simplification could make the expression simpler.

Ultimately, simplifying isn't just about removing complex elements, but also about making an expression easier to work with. This helps in mathematical operations like addition, subtraction, multiplication, and division and aids in deeper understanding.