Problem 22 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

x = 64.

Step 1: Understand the Equation

We are given the equation \( x^{2/3} = 16 \). Our task is to solve this equation for \( x \). This equation involves a rational exponent, 2/3, which implies we have to find a base number \( x \) such that when raised to the power of 2/3, the result is 16.

Step 2: Isolate the Exponent

Firstly, let's rewrite the equation so it is easier to handle. Instead of \( x^{2/3} \), we can think about this as \((x^{1/3})^2 = 16\). This step makes it clear that we need to deal with \( x^{1/3} \).

Step 3: Apply the Inverse Operation

To eliminate the power of two on \( x^{1/3} \), we take the square root of both sides. This gives us \( x^{1/3} = \sqrt{16} \). Since \( \sqrt{16} = 4 \), we have \( x^{1/3} = 4 \).

Step 4: Solve for x

Now, to remove the cube root, we cube both sides to solve for \( x \). Thus, we raise each side of the equation \( x^{1/3} = 4 \) to the power of 3: \((x^{1/3})^3 = 4^3 \), which simplifies to \( x = 64 \).

Key Concepts

Equation Solving

Equation solving is a fundamental concept in mathematics where the main goal is to find the unknowns. In our case, we are tasked with solving the equation \( x^{2/3} = 16 \). The solution involves working through a series of logical steps to manipulate and simplify the equation. Always begin by understanding what is being asked. Here, we aim to determine the value of \( x \) that satisfies the given equation. The first step is to understand the equation and identify any mathematical operations or concepts involved. Recognizing both rational exponents and how they affect the variable is crucial. After understanding the equation, the next task is to isolate the unknown variable on one side, allowing us to find its value easily.

Inverse Operations

Inverse operations are mathematical operations that can undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. In the realm of solving equations, using inverse operations is often the key to isolating a variable.

• In this exercise, to solve \( x^{1/3}=4 \), we cube both sides because the cube is the inverse operation of taking the cube root.
• Similarly, squaring or taking a square root are inverses, as shown when simplifying \((x^{1/3})^2=16\) by taking the square root to get \(x^{1/3}=4\).

To successfully apply inverse operations, recognize what operation has been performed on your variable. Then, apply the appropriate inverse to simplify the equation until the variable is isolated.

Exponents and Roots

Exponents and roots are closely connected concepts. An exponent indicates how many times a number should be multiplied by itself, whereas a root specifies which number, multiplied by itself a certain number of times, gives the original number.

• Rational exponents like \(2/3\) are a combination of both concepts: \(x^{2/3}\) means the cube root of \( x \) is squared.
• Simplifying expressions with rational exponents can involve understanding both powers and roots, as illustrated by converting \( x^{2/3} = 16 \) to \((x^{1/3})^2 = 16\).

For our equation, the technique requires recognizing how \( x^{1/3} \) represents the cube root and how subsequent steps like cubing can resolve the expression. Mastering these concepts is essential for handling equations involving rational exponents efficiently.