Problem 76 - Algebra and Trigonometry

Question

\(-\frac{1}{\sqrt{3}}\)

Step-by-Step Solution

Answer

-√3/3

Step 1: Understand the Problem

We analyze the given problem to determine what is being asked.

Step 2: Set Up the Solution

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\). Doing so will simplify the denominator, and making our expression more readable. We get: \(-\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\)

Step 3: Solve the Problem

Applying the relevant mathematical techniques, we work through the solution step by step.

Step 4: State the Result

\(-√3/3\)

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions can be a bit tricky at first, but with practice, it becomes straightforward. A radical expression contains a square root, cube root, or higher root. The key to simplifying them is to reduce the expression to its simplest form. This may involve factoring out perfect squares or other significant factors.

• First, identify if the number inside the radical can be factored into smaller numbers. For instance, the square root of 18 can be broken down into the square root of 9 (which is 3) and the square root of 2.
• Rewrite the expression as the product of the square roots, and then simplify by taking the square root of the perfect square. For our example, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\).

Mastering these steps can make complex expressions much more manageable. Don't rush, take your time to identify and extract perfect factors.

Fraction Simplification

Fraction simplification involves reducing the fraction to its lowest terms. This means dividing both the numerator and the denominator by their greatest common divisor (GCD).

• To begin, calculate the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this number to reduce the fraction.

In some cases, like when dealing with polynomials, simplification might also involve factoring and canceling out like terms.
For example, simplifying the fraction \(\frac{15}{35}\) can be done by dividing both numerator and denominator by 5. This results in \(\frac{3}{7}\).
Remember to always check if the new fraction can be simplified further.

Radicals

Radicals are mathematical symbols that indicate the root of a number. The most common radical is the square root, represented by the symbol \(\sqrt{}\). Radicals can also indicate higher roots, such as cube roots.Understanding radicals requires knowing the difference between a perfect square and a non-perfect square:

• A perfect square is a number that can be expressed as the product of an integer with itself, like \(16 = 4 \times 4\).
• For non-perfect squares, the square root is an irrational number, such as \(\sqrt{2}\).

Manipulating radicals in expressions often involves simplifying them by removing perfect square components, as explained in the first section. An important rule to remember is that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), which is valuable when simplifying expressions with radicals.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They follow specific rules and conventions, such as the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
To simplify algebraic expressions:

• First, expand any expressions with parentheses, ensuring to distribute any multiplied factors correctly.
• Combine like terms, which are terms that have the same variable raised to the same power.
• Perform any additional arithmetic to reduce the expression to its simplest form.

For instance, simplifying \(2(x + 3) + 4x\) involves distributing the 2 into the terms within the parentheses and then combining like terms, resulting in \(2x + 6 + 4x = 6x + 6\).
Understanding these principles helps with rationalizing denominators, as in the original exercise.