Problem 95 - Intermediate Algebra

Question

Consider the equations \(\sqrt{2 x}=4\) and \(\sqrt[3]{2 x}=4\) a. Explain the difference in solving these equations. b. Explain the similarity in solving these equations.

Step-by-Step Solution

Answer

Solving the first equation involves squaring, while the second involves cubing. Both require isolating \( x \) by dividing after removing the radical.

Step 1: Understanding Each Equation

The first equation is a square root equation, \( \sqrt{2x} = 4 \). It involves finding a value of \( x \) such that \( 2x \) raised to the power of 1/2 equals 4. The second equation is a cube root equation, \( \sqrt[3]{2x} = 4 \). It requires finding a value of \( x \) such that \( 2x \) raised to the power of 1/3 equals 4.

Step 1 for Equation \( \sqrt{2x} = 4 \)

Square both sides of the equation to remove the square root: \( (\sqrt{2x})^2 = 4^2 \). This simplifies to \( 2x = 16 \).

Step 2 for Equation \( \sqrt{2x} = 4 \)

Isolate \( x \) by dividing both sides by 2: \( x = \frac{16}{2} \). Therefore, \( x = 8 \).

Step 1 for Equation \( \sqrt[3]{2x} = 4 \)

Cube both sides of the equation to remove the cube root: \( (\sqrt[3]{2x})^3 = 4^3 \). This simplifies to \( 2x = 64 \).

Step 2 for Equation \( \sqrt[3]{2x} = 4 \)

Isolate \( x \) by dividing both sides by 2: \( x = \frac{64}{2} \). Therefore, \( x = 32 \).

Step 6: Difference in Solving

The main difference in solving these equations lies in the approach towards eliminating the radical: squaring versus cubing. The first equation involves squaring both sides to eliminate a square root, whereas the second involves cubing to eliminate a cube root.

Step 7: Similarity in Solving

The similarity in solving these equations is the method of isolating \( x \) after removing the radical. For both equations, after applying the appropriate power, we simplify to \( 2x = \text{value} \), and then solve for \( x \) by dividing by 2.

Key Concepts

Square Root Equations

Square root equations involve the radical symbol \( \sqrt{} \), which means you are dealing with a number raised to the power of \( \frac{1}{2} \). For example, in the equation \( \sqrt{2x} = 4 \), your goal is to eliminate the square root. To do this, you need to reverse the operation of taking a square root—by squaring both sides of the equation.

• First, square both sides: \( (\sqrt{2x})^2 = 4^2 \).
• This results in the equation \( 2x = 16 \).
• Finally, solve for \( x \) by dividing both sides by 2: \( x = 8 \).

After squaring, the equation turns into a more straightforward algebraic equation, making it easier to isolate and solve for \( x \). By understanding this process, you can tackle any square root equation methodically.

Cube Root Equations

Cube root equations incorporate the cube root symbol \( \sqrt[3]{} \), which represents a number raised to the power of \( \frac{1}{3} \). For instance, the equation \( \sqrt[3]{2x} = 4 \) requires a method to eliminate the cube root. This method involves the opposite operation of taking a cube root, which is cubing both sides of the equation.

• Start by cubing both sides: \( (\sqrt[3]{2x})^3 = 4^3 \).
• What you get is \( 2x = 64 \).
• Next, isolate \( x \) by dividing both sides by 2: \( x = 32 \).

After cubing, the equation simplifies into a form that is easier to handle, allowing you to solve for \( x \) efficiently. Knowing this procedure is beneficial in successfully solving cube root equations with confidence.

Solving Equations

The solution of equations typically requires transforming the problem into one or more steps that simplify the equation. Whether dealing with square root or cube root equations, the strategy always involves the same basic steps: eliminating the radical and isolating the variable \( x \).

The techniques for solving:

• Identify the Radical: Determine whether you're working with a square root or a cube root.
• Eliminate the Radical: Use the opposite operation (squaring or cubing) to remove the radical.
• Isolate the Variable: Simplify the resulting algebraic equation and solve for \(x\) using basic algebraic operations.

Though the operations differ—squaring for square roots and cubing for cube roots—the approach to solving these problems is quite similar. Ultimately, understanding how to methodically eliminate the radical and isolate \( x \) will guide you in solving many types of radical equations.