Problem 25 - A Graphical Approach to College Algebra

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$(\sqrt[4]{y})^{-3}$$

Step-by-Step Solution

Answer

The expression rewritten with a positive rational exponent is \(\frac{1}{y^{\frac{3}{4}}}\).

Step 1: Understand the Problem

You are given the expression \((\sqrt[4]{y})^{-3}\). This expression needs to be rewritten using positive rational exponents. The key is to remember that the radical and the negative exponent have specific ways to be converted into rational exponents.

Step 2: Identify the Radical Expression

The expression \(\sqrt[4]{y}\) can be rewritten using rational exponents. A radical in the form of \(\sqrt[n]{a}\) is equivalent to \(a^{1/n}\). Hence, \(\sqrt[4]{y}\) can be rewritten as \(y^{1/4}\).

Step 3: Apply the Negative Exponent

The entire expression \((y^{1/4})^{-3}\) now involves applying the negative exponent. Recall that a negative exponent \(a^{-m}\) can be rewritten as \(1/a^m\). So, \((y^{1/4})^{-3}\) becomes \((y^{1/4})^{3}\) in the denominator.

Step 4: Multiply the Exponents

When you have \((a^m)^n\), you multiply the exponents: \(a^{m \cdot n}\). Therefore, \((y^{1/4})^{-3}\) becomes \(y^{-(3 \cdot \frac{1}{4})}\). Simplify the exponent: \(-3 \cdot \frac{1}{4} = -\frac{3}{4}\). Thus, the expression is now \(y^{-\frac{3}{4}}\).

Step 5: Write with a Positive Exponent

Convert \(y^{-\frac{3}{4}}\) into a form with a positive exponent by moving it to the denominator: \(\frac{1}{y^{\frac{3}{4}}}\). This is the expression rewritten with a positive rational exponent.

Key Concepts

Understanding Negative Exponents

Negative exponents might seem tricky at first, but they play a crucial role in simplifying expressions. Simply put, a negative exponent indicates how many times to divide by the number, as opposed to multiplying. For example:

• An expression like \(a^{-n}\) should be translated into \(\frac{1}{a^n}\).
• This essentially means you're flipping the position of the base from the numerator to the denominator or vice versa, depending on where it starts.

This approach significantly simplifies expressions and helps in managing calculations in a more systematic manner. When you use negative exponents during calculations:

• Think of them as a nifty tool to transition numbers and variables between being multiplied and divided.
• Remember to always convert negative exponents into positive ones for simpler interpretations and further calculations.

Grasping this concept helps in deeper mathematical understanding and application of other algebraic rules.

Exploring Radical Expressions

Radical expressions involve roots and can be expressed with or without the radical sign \(\sqrt{\phantom{x}}\). In many instances, expressing radicals using rational exponents offers more flexibility. Here's how:

• A radical expression like \(\sqrt[n]{a}\) can be rewritten as \(a^{1/n}\).
• This translation helps when performing operations involving radicals, especially when simplifying more complex expressions.

Converting radicals into rational exponents is particularly beneficial in algebra:

• It allows you to leverage properties of exponents, such as multiplying and dividing, simplifying the processes.
• During this transformation, the exponent represents the fraction where the numerator is the power and the denominator is the root.

By expressing radical terms as rational exponents, you gain the flexibility to easily manipulate them, especially when combined with other algebraic terms.

Rational Numbers Deeper Dive

Rational numbers are foundational in mathematics. They are numbers that can be expressed as the fraction of two integers \(\frac{a}{b}\), where \(b eq 0\). This makes them incredibly versatile and useful in algebra.

• The most prominent feature of rational numbers is the ability to express divisions traditionally encountered in math.
• They include both positive, negative fractions, and whole numbers, which makes them comprehensive in covering all non-decimal numbers.

Working with rational numbers allows us to:

• Transition smoothly between fractions and decimals when needed for calculations.
• Provide exact values for roots and powers, especially when converting complexities involving radicals.
• Ease solving equations through common denominators, aiding in simplifying algebraic expressions.

In algebra, understanding how to leverage rational numbers, especially when they appear as exponents, is critical. This is because they enhance both flexibility and clarity when dealing with equations and expressions.