Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation by hand. Do not use a calculator. $$x^{2 / 3}=16$$

Step-by-Step Solution

Answer

The solution is \( x = 64 \).

Step 1: Understand the Equation

The equation presented is \( x^{2/3} = 16 \). This is asking for the value of \( x \) such that, when raised to the power of \( \frac{2}{3} \), the result is 16.

Step 2: Isolate the Base Term

To solve for \( x \), we first need to eliminate the exponent. We can do this by raising both sides of the equation to the power of the reciprocal of \( \frac{2}{3} \). The reciprocal is \( \frac{3}{2} \).

Step 3: Raise Both Sides to the Power \( \frac{3}{2} \)

Raise both sides of the equation to the power of \( \frac{3}{2} \). This gives us: \[(x^{2/3})^{3/2} = 16^{3/2}\] which simplifies on the left to \( x \) because \((2/3) \times (3/2) = 1\).

Step 4: Calculate the Right Side

Now calculate \( 16^{3/2} \). First, find \( 16^{1/2} \), which is the square root of 16, giving 4. Then calculate \( 4^3 = 64 \). So, \( 16^{3/2} = 64 \).

Step 5: State the Solution

The calculations show that \( x = 64 \). Thus, the solution to the equation \( x^{2/3} = 16 \) is \( x = 64 \).

Key Concepts

Rational Exponents

Rational exponents extend the idea of powers to include fractions. They allow us to express roots as exponents, making operations more flexible. Typically, the expression \( a^{m/n} \) signifies the \( n \)-th root of \( a^m \). For example, \( x^{2/3} \) can be interpreted as the cube root of \( x^2 \). This helps in simplifying expressions and solving equations more efficiently.

They follow the same rules as integer exponents, making it simpler to work with complex expressions.

• Multiplication: \( a^{m/n} \times a^{p/q} = a^{(mq+np)/nq} \)
• Division: \( a^{m/n} \div a^{p/q} = a^{(mq-np)/nq} \)
• Powers of a power: \( (a^{m/n})^{p/q} = a^{mp/nq} \)

Understanding rational exponents is crucial as they are frequently used across various branches of mathematics including algebra, calculus, and beyond.

Solving Equations

Solving equations is all about finding the unknown that makes the equation true. When dealing with equations involving rational exponents, full understanding of exponent rules is key.

To solve equations like \( x^{2/3} = 16 \), the strategy involves isolating the variable. You achieve this by utilizing the property of inverse operations. First, raise both sides of the equation by the reciprocal of the exponent on \( x \), which in this case is \( 3/2 \).

• This simplification step involves evaluating \( 16^{3/2} \) using knowledge of exponents, calculating \( 16^{1/2} \) first which is 4, then raising it to the third power to get 64.

This method of isolating and calculating helps to gradually break down the problem into manageable parts, leading to our solution: \( x = 64 \). Practicing this approach with different equations will strengthen your problem-solving skills.

Mathematical Notation

Mathematical notation is a universal language used to represent numbers, variables, functions, and operations concisely. Understanding notation is essential for interpreting and solving mathematical problems.

With rational exponents like \( x^{2/3} \), the notation signifies a power and root, making it easier to manipulate algebraic expressions. Each component of the notation holds a specific meaning:

• \( x \) is the base, which is the number being acted upon.
• \( 2 \) is the numerator of the exponent, indicating that \( x \) is squared.
• \( 3 \) is the denominator, indicating that the cube root is taken.

This leveling up of understanding notation eliminates confusion and equips students with the skills to interpret complex expressions effectively. As mathematics builds on these fundamentals, mastering notation paves the way for success in advanced topics.