Problem 55 - Precalculus: Functions and Graphs

Question

Rewrite the expression using a radical. (a) \(\quad 8-y^{1 / 3}\) (b) \((8-y)^{13}\)

Step-by-Step Solution

Answer

(a) \(8 - \sqrt[3]{y}\); (b) No change.

Step 1: Identify the Expression (a)

The given expression is \(8 - y^{1/3}\). This expression involves a fractional exponent, which we need to convert to a radical form.

Step 2: Apply Radical Definition to (a)

Recall that the expression \(y^{1/3}\) is equivalent to the cube root of \(y\), which can be written as \(\sqrt[3]{y}\). Thus, the expression becomes \(8 - \sqrt[3]{y}\).

Step 3: Confirm Radical Conversion (a) Correctness

Check that \(\sqrt[3]{y}\) and \(y^{1/3}\) represent the same quantity, confirming that the expression \(8 - \sqrt[3]{y}\) is correctly written in radical form.

Step 4: Identify the Expression (b)

The given expression is \((8-y)^{13}\). This expression does not have a fractional exponent, therefore, it is not readily converted into a radical form.

Step 5: Evaluate Possibility for Radical Conversion (b)

Since the exponent is a whole number, \((8-y)^{13}\) does not need (or qualify for) rewriting into a radical. Expressions with whole exponents are typically left as-is.

Key Concepts

Fractional Exponents

Fractional exponents are a way of expressing powers and roots in a compact form. When you see an exponent like \( y^{1/n} \), it is telling you to take the nth root of \( y \). This is an essential idea to understand when working with algebra and precalculus problems. Fractional exponents can be daunting at first, but they simplify expressions involving roots and powers.

Here are some points to remember about fractional exponents:

• \( y^{1/2} \) means the square root of y, often written as \( \sqrt{y} \).
• \( y^{1/3} \) means the cube root of y, also written as \( \sqrt[3]{y} \).
• In general, \( y^{m/n} \) translates to the nth root of \( y^m \), \( \sqrt[n]{y^m} \).

When you see an expression like \( y^{1/3} \) in the exercise, it can be translated directly into a radical, \( \sqrt[3]{y} \). Understanding this conversion helps you manipulate and simplify expressions correctly.

Radical Conversion

Radical conversion is the process of rewriting expressions that contain fractional exponents into their radical forms. This conversion is pivotal in making complex expressions more intuitive and sometimes easier to handle.

• For each fractional exponent, identify the root it represents. For example, \( y^{1/3} \) is equivalent to the cube root \( \sqrt[3]{y} \).
• Converting to radicals can make it easier to visualize and understand the type of root involved, especially when dealing with multiple operations.

In the exercise, converting \( y^{1/3} \) to \( \sqrt[3]{y} \) allowed us to simplify the given expression. It demonstrates how radical forms can often be a simpler or more recognizable way of writing expressions that involve roots. This type of conversion does not change the value of the expression, merely its form.

Precalculus

Precalculus is a course that prepares students for calculus by covering various mathematical concepts, including algebra, trigonometry, and the introduction to limits and derivatives. Radical expressions and fractional exponents are common topics in precalculus, as they lay the groundwork for understanding more advanced calculus concepts. Precalculus helps students:

• Master algebraic manipulations and understand the relationship between fractional and whole number exponents.
• Understand the significance of radicals and their conversion from fractional exponents.
• Develop critical thinking and problem-solving skills necessary for calculus.
• Grasp the importance of converting expressions to different forms for simplification and comparison.

In the context of precalculus, knowledge of fractional exponents and radicals plays a vital role in handling polynomial equations, exponential functions, and even logarithms. These concepts are foundational for success in calculus, where you will deal with increasingly complex expressions.