Problem 1 - A Graphical Approach to College Algebra

Question

Match the rational exponent expression in Exercises \(1-8\) with the equivalent radical expression in \(A-H\). Assume that \(x \neq 0\). A. \(\frac{3}{\sqrt[3]{x}}\) B. \(-3 \sqrt[3]{x}\) C. \(\frac{1}{\sqrt[3]{3 x}}\) D. \(\frac{-3}{\sqrt[3]{x}}\) E. \(3 \sqrt[3]{x}\) F. \(\sqrt[3]{-3 x}\) G. \(\sqrt[3]{3 x}\) H. \(\frac{1}{\sqrt[3]{-3 x}}\) $$(-3 x)^{1 / 3}$$

Step-by-Step Solution

Answer

Match the expression \((-3x)^{1/3}\) with \(F. \sqrt[3]{-3 x}\).

Step 1: Understand the Rational Exponent

The given expression is \((-3x)^{1/3}\). This expression involves a base of \(-3x\) and an exponent of \(1/3\). In terms of radical notation, this exponent signifies a cube root.

Step 2: Convert to Radical Notation

Convert the expression \((-3x)^{1/3}\) into radical form. A rational exponent of \(1/3\) means "cube root." Thus, \((-3x)^{1/3}\) translates to \(\sqrt[3]{-3x}\).

Step 3: Match with Provided Options

Now, compare the expression \(\sqrt[3]{-3x}\) to the list of options provided (A to H). The correct match is option \(F\), which is \(\sqrt[3]{-3x}\).

Key Concepts

Radical expressions

Radical expressions are mathematical expressions that involve roots. These can be square roots, cube roots, or other types of roots. The root itself is symbolized by a radical sign, \( \sqrt{} \), often with an index to denote the type of root; for example, \( \sqrt[3]{} \) for a cube root. When you encounter a radical expression, you're essentially looking at the problem of finding a number which, raised to a certain power, will give the original number inside the radical.

There are a few key things to remember about radical expressions:

• The index is the small number written just outside and above the radical sign. If no number is written, it is assumed to be 2, indicating a square root.
• The expression inside the radical sign is called the radicand. In the expression \( \sqrt[3]{-3x} \), \(-3x\) is the radicand.
• Radical expressions can be simplified by factoring the radicand into terms that are perfect powers of the root. For example, \( \sqrt{36} \) can be simplified to \( 6 \).
• Remember to pay attention to negative signs, especially with square roots, since square roots of negative numbers involve imaginary numbers. However, cube roots and other odd roots can have negative results.

Understanding radicals helps us appreciate the simplicity they offer in representing complex roots while solving equations or simplifying expressions.

Cube root

The cube root of a number is a special radical expression where the index is 3. This means that you need to find a number which, multiplied by itself three times, gives the original number. Cube roots are denoted as \( \sqrt[3]{} \). For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).

Let's understand some important points about cube roots:

• An important property of cube roots is that they can be taken of negative numbers, leading to negative results. For example, \( \sqrt[3]{-8} = -2 \) because \( -2 \times -2 \times -2 = -8 \).
• When dealing with expressions like \( (-3x)^{1/3} \), we identify this as \( \sqrt[3]{-3x} \) by converting the rational exponent to a cube root.
• Unlike square roots, cube roots of perfect cubes are always integers. For example, \( \sqrt[3]{27} = 3 \) since \( 3^3 = 27 \).

Cube roots help simplify expressions and solve algebraic equations where the variable is raised to the third power. Alongside their use in everyday mathematics, they have applications in physics, engineering, and computer science when modeling three-dimensional problems.

Notation conversion

In mathematics, converting between different notations is crucial for solving problems effectively. When it comes to radical expressions and rational exponents, mastering notation conversion can make complex expressions easier to manage. Rational exponents and radicals are closely related and can be interchangeably expressed.

Here are essential points about notation conversion:

• Rational exponents are another way to denote roots. The expression \( a^{1/n} \) is the same as \( \sqrt[n]{a} \). This means you can convert expressions back and forth depending on which notation simplifies the problem easily.
• The general rule for converting from rational exponents to radical notation is given by \( a^{m/n} = \sqrt[n]{a^m} \). For example, \( 9^{1/2} = \sqrt{9} \).
• In the problem scenario, \( (-3x)^{1/3} \) is converted into \( \sqrt[3]{-3x} \), indicating the cube root of the expression inside.
• Understanding these conversions helps in algebraic manipulation, ensuring that expressions can be rewritten in the most useful or necessary form for solving equations or simplifying mathematical models.

By learning notation conversion between radicals and exponents, students can solve a wider variety of problems with greater flexibility and insight.