Problem 124 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \frac{3 t-1}{\sqrt{3 t}+1} $$

Step-by-Step Solution

Answer

The fully simplified expression is \( \frac{3t\sqrt{3t} - 3t - \sqrt{3t} + 1}{3t - 1} \).

Step 1: Identify the need to rationalize the denominator

The expression \( \frac{3t - 1}{\sqrt{3t} + 1} \) involves a denominator with a radical. To simplify, we need to rationalize the denominator by removing the square root.

Step 2: Multiply by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{3t} - 1 \). This gives us:\[\frac{(3t - 1)(\sqrt{3t} - 1)}{(\sqrt{3t} + 1)(\sqrt{3t} - 1)}\]

Step 3: Simplify the denominator using the difference of squares

The denominator becomes a difference of squares: \[(\sqrt{3t} + 1)(\sqrt{3t} - 1) = (\sqrt{3t})^2 - 1^2 = 3t - 1\]

Step 4: Expand the numerator

Expand the numerator by distributing:\[(3t - 1)(\sqrt{3t} - 1) = 3t(\sqrt{3t}) - 3t - 1(\sqrt{3t}) + 1\]Simplifying further, we get:\[3t\sqrt{3t} - 3t - \sqrt{3t} + 1\]

Step 5: Combine the simplified fraction

Include both the simplified numerator and denominator into a single fraction:\[\frac{3t \sqrt{3t} - 3t - \sqrt{3t} + 1}{3t - 1}\]

Step 6: Check for common factors

Since there are no common factors in the numerator and denominator that can be simplified further, the expression is fully simplified.

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include a root symbol, typically a square root or a cube root. In the given exercise, the term \( \sqrt{3t} \) is a radical expression, representing the square root of \( 3t \). Radical expressions can often become unwieldy, especially when they appear in the denominators of fractions.

To deal with radical expressions, it is often desirable to simplify them. Simplification can involve combining like terms or rationalizing. Rationalizing a denominator that contains a radical is particularly important because most mathematicians prefer expressions without radicals in the denominator.

When simplifying radical expressions, keep these key tips in mind:

• Try to factor out perfect square factors.
• Use identities like \( a^2 - b^2 \) to simplify complex expressions.
• Remember that you can often use the conjugate to eliminate radicals in denominators.

Difference of Squares

The difference of squares is a crucial algebraic identity that comes in handy when simplifying expressions. The identity is \( a^2 - b^2 = (a+b)(a-b) \). This identity can be a powerful tool, especially when rationalizing denominators that contain radicals.

In the exercise, when we used the conjugate to rationalize the denominator \((\sqrt{3t} + 1)(\sqrt{3t} - 1)\), we took advantage of this identity. The expression simplifies to \( \sqrt{3t}^2 - 1^2 = 3t - 1 \), neatly removing the radical from the denominator.

Here’s why the difference of squares is so helpful:

• It efficiently removes square roots from expressions.
• It simplifies computations by turning complicated terms into simpler products.
• Remembering this identity can save time and avoid common mistakes.

Knowing how and when to apply the difference of squares can greatly simplify your mathematical work.

Conjugate Multiplication

Conjugate multiplication is an effective technique to simplify expressions, particularly when the goal is to rationalize denominators with radicals. The conjugate of an expression is formed by changing the sign between two terms in a binomial. For example, the conjugate of \( \sqrt{3t} + 1 \) is \( \sqrt{3t} - 1 \).

By multiplying the numerator and the denominator of a fractional expression by the conjugate of the denominator, you eliminate the radicals. This process is termed as conjugate multiplication.

In our example,

• The original denominator \( \sqrt{3t} + 1 \) simplifies to \( 3t - 1 \) through conjugate multiplication.
• The numerator also transforms, although the goal is primarily to manage the denominator.

Remember these points when using conjugate multiplication:

• Ensure that both the numerator and denominator are multiplied by the conjugate to maintain expression equality.
• This method often pairs with the difference of squares to achieve a clean, simplified form.
• Practicing this technique enhances simplification skills, particularly for complex fraction problems.

Mastering conjugate multiplication will make working with radical expressions considerably easier.