Problem 49 - College Algebra

Question

For the following exercises, simplify each expression. $$ \frac{\sqrt{8}}{1-\sqrt{3 x}} $$

Step-by-Step Solution

Answer

\(\frac{2\sqrt{2} + 2\sqrt{6x}}{1 - 3x}\) is the simplified form.

Step 1: Recognize the Objective

We want to simplify the given expression \( \frac{\sqrt{8}}{1-\sqrt{3x}} \). Our main task is to make the expression more straightforward or easier to understand or use, often by eliminating radicals or complex fractions in the denominator.

Step 2: Simplify the Numerator

The numerator \( \sqrt{8} \) can be simplified by recognizing it as \( \sqrt{4 \times 2} \), which becomes \( \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \). Now the expression updates to \( \frac{2\sqrt{2}}{1-\sqrt{3x}} \).

Step 3: Rationalize the Denominator

To simplify further, we need to rationalize the denominator. This involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 - \sqrt{3x}\) is \(1 + \sqrt{3x}\). So we multiply by \(\frac{1+\sqrt{3x}}{1+\sqrt{3x}}\).

Step 4: Expand the Numerator

The numerator becomes \((2\sqrt{2})(1+\sqrt{3x}) = 2\sqrt{2} + 2\sqrt{2} \cdot \sqrt{3x} = 2\sqrt{2} + 2\sqrt{6x}\).

Step 5: Simplify the Denominator

The denominator \((1-\sqrt{3x})(1+\sqrt{3x})\) simplifies using the difference of squares formula: \(1^2-(\sqrt{3x})^2 = 1 - 3x\).

Step 6: Write the Final Expression

Now that we have simplified both the numerator and the denominator, the expression becomes \(\frac{2\sqrt{2} + 2\sqrt{6x}}{1 - 3x}\). This is the simplified form of the original expression.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a crucial step in simplifying expressions that involve radicals. The goal of rationalization is to eliminate any radical expression from the denominator of a fraction, making it easier to work with or evaluate. To achieve this, you multiply both the numerator and the denominator of the fraction by the conjugate of the denominator.

The conjugate of a binomial expression like \( 1 - \sqrt{3x} \) is \( 1 + \sqrt{3x} \). Multiplying by the conjugate changes the denominator into a difference of squares. This often results in a rational, non-radical expression.

When applied to the example \( \frac{\sqrt{8}}{1-\sqrt{3x}} \), multiplying by \( \frac{1+\sqrt{3x}}{1+\sqrt{3x}} \) gives a rationalized denominator of \( 1 - (\sqrt{3x})^2 = 1 - 3x \). This process simplifies the fraction to make calculations more straightforward.

Radical Expressions

Radical expressions are those which involve roots, such as square roots or cube roots. In the expression \( \sqrt{8}\), we are dealing with a square root. Simplifying radical expressions involves factoring the radicand (the number under the root symbol) into its prime factors and extracting square factors.

For the radical \( \sqrt{8} \), this can be broken down as \( 8 = 4 \times 2\). Since \( 4 \) is a perfect square,\( \sqrt{8} \) simplifies to \( \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \).

The simplified radical expression is then used in further calculations, like in rationalizing denominators. Understanding these simplifications can greatly ease the work involved in algebraic manipulation.

Difference of Squares

The difference of squares is a specific algebraic formula used often in simplifying expressions. It takes the form \( a^2 - b^2 \), which can be broken down into \( (a-b)(a+b) \). This is extremely useful when rationalizing a denominator in radical expressions.

In the given problem involving \( 1 - \sqrt{3x} \), the conjugate is used to transform the expression into a difference of squares. More specifically, multiplying \( (1 - \sqrt{3x})(1 + \sqrt{3x}) \) results in \( 1^2 - (\sqrt{3x})^2 = 1 - 3x \), which is a simpler, polynomial expression.

Recognizing and applying the difference of squares can significantly simplify complex algebraic problems, allowing further simplification or calculation to proceed with easier, rational expressions.