Problem 26 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ (x-4)^{3 / 4} $$

Step-by-Step Solution

Answer

\( \sqrt[4]{(x-4)^3} \) is the expression in radical notation.

Step 1: Understanding the Power Fraction

Recognize that the exponent \( \frac{3}{4} \) can be split into two components: a cube power (numerator) and a fourth root (denominator). This means \( (x-4)^{3/4} \) denotes both a cube and a root.

Step 2: Rewrite as a Radical

Convert the expression \( (x-4)^{3/4} \) into radical notation: use the denominator of the fraction as a root and the numerator as the power, resulting in \( \sqrt[4]{(x-4)^3} \).

Step 3: Apply Radical Notation

The expression in radical notation is \( \sqrt[4]{(x-4)^3} \). This is the simplified expression using radical notation.

Key Concepts

Exponents

Exponents, also known as powers, represent how many times a number or expression is multiplied by itself. In the expression \((x-4)^{3/4}\), the portion outside the bracket \(3/4\) is the exponent. Exponents play a vital role in mathematical operations, especially with complex expressions.

An exponent is usually indicated by a small number placed to the upper right of the base value. Here are some key points about exponents:

• The exponent 1 means the number is used once, essentially the number itself.
• An exponent of 2 indicates squaring the base, which is the base multiplied by itself once.
• Zero as an exponent equals 1, regardless of the base (as long as the base is not zero).

Exponents can also be fractions, which we'll explore in the next section. Understanding how to read and manipulate exponents is crucial for success in algebra and beyond.

Fractional Exponents

Fractional exponents are a type of exponent expressed as fractions. Unlike integer exponents, fractional exponents denote roots, as well as powers. For example, \((x-4)^{3/4}\) includes a fractional exponent. The numerator of the fraction indicates the power (cubing in this case), while the denominator represents the root (the fourth root).

By rewriting \((x-4)^{3/4}\) as \(\sqrt[4]{(x-4)^3}\), we transform the expression into its radical form, which is another way to express fractional exponents. Here’s how you can better understand this concept:

• Numerator as Power: The numerator (3) indicates the exponent to which the base is raised, essentially \((x-4)^3\).
• Denominator as Root: The denominator (4) refers to the root that is applied to the resulting expression, here it's the fourth root \(\sqrt[4]{(x-4)^3}\).

This dual interpretation makes fractional exponents very versatile in simplifying and solving algebraic expressions.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in a more straightforward or more understandable form. In algebra, this often means removing any roots, exponents, or extraneous numbers when possible. Simplifying makes an expression easier to evaluate or use in further calculations. For example, converting \((x-4)^{3/4}\) into its radical notation \(\sqrt[4]{(x-4)^3}\) streamlines the original expression.

Here are some steps and tips for simplifying expressions:

• Identify the components: Look for exponents, roots, or fractions that can be combined or rewritten.
• Apply arithmetic operations: Use addition, subtraction, multiplication, and division to consolidate terms where applicable.
• Convert to radical notation: If dealing with fractional exponents, rewrite them using radical notation for a clear understanding.

Remember, the goal is to make expressions as uncomplicated as possible without altering their value. This will facilitate easier problem resolution and allow for more effective and efficient manipulation in algebraic practices.