Problem 41 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ (5 r+14)^{1 / 3}=4 $$

Step-by-Step Solution

Answer

The solution is \(r = 10\).

Step 1: Understand the Equation

The given equation is \((5r + 14)^{1/3} = 4\). This equation involves a cube root, as indicated by the exponent \(\frac{1}{3}\). To solve for \(r\), we need to eliminate the cube root by cubing both sides of the equation.

Step 2: Eliminate the Cube Root

Cube both sides of the equation to get rid of the cube root on the left-hand side:\[((5r + 14)^{1/3})^3 = 4^3\] Simplifying both sides, we obtain:\[ 5r + 14 = 64 \]

Step 3: Isolate the Variable Term

Subtract 14 from both sides to isolate the term containing \(r\):\[5r + 14 - 14 = 64 - 14\]This simplifies to:\[5r = 50\]

Step 4: Solve for the Variable

Divide both sides of the equation by 5 to solve for \(r\):\[\frac{5r}{5} = \frac{50}{5}\]Therefore, we find:\[r = 10\]

Key Concepts

Solving Equations

In algebra, solving equations is like solving a puzzle. You have all the pieces, but you need to place them correctly to see the full picture. An equation comprises numbers, variables (like \( r \)), and operations (like addition or multiplication). The goal is to find the value of the variable that makes the equation true. To do this:

• We typically need to perform the same operation on both sides of the equation. This keeps the balance, just like balancing scales.
• Each operation reverses the effect of another. Adding undoes subtracting, multiplying undoes dividing, and vice versa.
• By using these operations, we can isolate the variable and solve the equation step by step.

Tackling each step systematically helps in understanding and solving the problem accurately. In our case, solving for \( r \) involved using cube roots, which leads us to the next important concept.

Cube Roots

A cube root is a number that, when multiplied by itself two more times, results in the original number. In simpler terms, if you cube a number (multiply it by itself twice), and then find its cube root, you get back to the original number. For instance:

• If \( 2^3 = 8 \), then the cube root of 8 is 2.
• Cubing a number is denoted as raising to the power of 3.
• The cube root of a number is denoted by the exponent \(1/3\).

In our equation, the cube root \((5r + 14)^{1/3} = 4\) means that when 5r + 14 is cubed, it equals 64. To remove a cube root, we cube both sides of the equation. This eliminates the root, making it easier to solve for the variable. Cubing both sides ensures we stay accurate and maintain the equation's balance.

Isolating Variables

Isolating variables is an essential skill in algebra. We aim to get the variable alone on one side of the equation to find its value. Here's how you can systematically approach isolating variables:

• Identify any additional numbers or operations attached to the variable.
• Use inverse operations to systematically remove these attachments.
• Every step must be mirrored on both sides of the equation to maintain equality.

In our problem, once we eliminated the cube root, our next task was to isolate \( r \). We first subtracted 14 from both sides (reversing the addition), resulting in \( 5r = 50 \). Finally, dividing both sides by 5 (the coefficient of \( r \)), isolated \( r \) and solved the equation: \( r = 10 \). This process is logical and ensures the integrity of the equation throughout your calculations.