Problem 72 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$(3 x+7)^{1 / 3}-(4 x+2)^{1 / 3}=0$$

Step-by-Step Solution

Answer

x = 5

Step 1: Isolate the Cubic Roots

Add \( (4x + 2)^{1/3} \) to both sides of the equation to isolate the cubic roots: \[ (3x + 7)^{1/3} = (4x + 2)^{1/3} \]

Step 2: Cube Both Sides

Cube both sides of the equation to eliminate the cube roots: \[ \big[(3x + 7)^{1/3}\big]^3 = \big[(4x + 2)^{1/3}\big]^3 \]This simplifies to: \[ 3x + 7 = 4x + 2 \]

Step 3: Solve for x

Rearrange the equation to isolate \ x \ on one side: \[ 3x + 7 - 3x = 4x + 2 - 3x \]Simplify the equation: \[ 7 = x + 2 \]Subtract 2 from both sides: \[ x = 5 \]

Step 4: Verify the Solution

Plug \ x = 5 \ back into the original equation to verify the solution: \[ (3(5) + 7)^{1/3} - (4(5) + 2)^{1/3} = 0 \]Simplify each term: \[ (15 + 7)^{1/3} - (20 + 2)^{1/3} = 0 \] \[ 22^{1/3} - 22^{1/3} = 0 \]Since the equation holds true, \ x = 5 \ is the correct solution.

Key Concepts

Cubic Roots

A cubic root, also known as the cube root, is a number that, when multiplied by itself three times, gives the original number. Mathematically, the cube root of a number y is written as \(y^{1/3}\). For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). Cube roots are the inverse operations of cubing a number. They are useful for solving equations where variables are under a cube root.

Cubing Both Sides

Cubing both sides of an equation is a technique used to eliminate cube roots. This is particularly useful when dealing with equations where variables are under cube roots, as it simplifies the equation. For example, in the equation \((3x+7)^{1/3} = (4x+2)^{1/3}\), you can cube both sides to remove the cube roots:

• Take the cube of \((3x + 7)^{1/3}\), which gives you \((3x + 7)\)
• Take the cube of \((4x + 2)^{1/3}\), which results in \((4x + 2)\)

After cubing both sides, the equation simplifies to \(3x + 7 = 4x + 2\), which is easier to solve. This technique is a powerful tool for transforming complex equations into simpler, more manageable forms.

Verifying Solutions

After solving an equation, it's important to verify the solution. This step ensures that your solution is correct and satisfies the original equation. To verify a solution:

• Substitute the solution back into the original equation.
• Simplify both sides to check if they are equal.

For example, if you solved \(x = 5\) for the original equation \((3x + 7)^{1/3} - (4x + 2)^{1/3} = 0\), you would substitute \(x = 5\) back into it:

• Simplify \((3(5) + 7)^{1/3}\) and \((4(5) + 2)^{1/3}\).
• Check if both sides equal zero after simplification.

If they do, your solution is verified and correct. Verifying helps catch any mistakes that might have been made during the solving process.

Isolating Variables

Isolating variables is a key step in solving equations. It involves rearranging the equation so that the variable you’re solving for is by itself on one side of the equation. This helps to clearly identify the value of the variable. In the given equation \(3x + 7 = 4x + 2\), you can isolate \(x\) by:

• Subtracting \(3x\) from both sides: \(4x + 2 - 3x = 3x + 7 - 3x\), which simplifies to \(x + 2 = 7\).
• Subtracting 2 from both sides: \(x + 2 - 2 = 7 - 2\), simplifying to \(x = 5\).

By following these steps, we successfully isolate \(x\) and find its value. Isolating the variable is a fundamental technique in algebra that makes solving equations straightforward and efficient.