Problem 41 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(r = 10\).

Step 1: Understand the Equation

The given equation is \((5r + 14)^{1/3} = 4\). The goal is to solve for \(r\). Notice that the equation involves a cube root on the left side.

Step 2: Eliminate the Cube Root

To eliminate the cube root, cube both sides of the equation to get rid of the fractional exponent. So we have:\[((5r + 14)^{1/3})^3 = 4^3\]Which simplifies to:\[5r + 14 = 64\]

Step 3: Isolate the Term with r

Subtract 14 from both sides of the equation to isolate the term with \(r\):\[5r + 14 - 14 = 64 - 14\]This gives us:\[5r = 50\]

Step 4: Solve for r

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

Step 5: Verify the Solution

To verify, substitute \(r = 10\) back into the original equation:\[(5(10) + 14)^{1/3} = 4\]Simplify inside the parentheses:\[(50 + 14)^{1/3} = 4\]\[64^{1/3} = 4\]Since \(4\) is indeed equal to \(4\), the solution is correct.

Key Concepts

Cube Root

Understanding cube roots is a fundamental part of solving equations involving exponents. A cube root of a number is what you multiply by itself twice to get that number. For example, the cube root of 64 is 4 because when you multiply 4 by itself twice (i.e., 4 \( \times \) 4 \( \times \) 4), the result is 64.

Cube roots are represented by the symbol \( \sqrt[3]{} \) or the exponent \( ^{1/3} \). In algebraic equations, cube roots often appear in the form \( (something)^{1/3} = number \). To simplify and solve these equations, you'll typically need to eliminate the cube root by performing the opposite operation, which is cubing both sides.

• Example: To remove \( (5r + 14)^{1/3} \), cube both sides to obtain \( 5r + 14 \).
• By eliminating the cube root in this manner, it simplifies the equation and makes isolating the variable easier.

Isolation Technique

The isolation technique is a fundamental skill in solving equations, especially when dealing with algebraic expressions. It involves manipulating the equation to get the variable by itself on one side of the equation. In our step-by-step solution, after cubing the equation to remove the cube root, the equation reduced to \(5r + 14 = 64\).

The isolation process involved the following steps:

• Subtracting 14 from both sides gave \(5r = 50\).
• Then, dividing by 5 gave \(r = 10\).

The goal of isolation is always to have one side of the equation as "the variable equals". This method requires reversing mathematical operations by following the order of operations in reverse.
This allows you to effectively and accurately find the solution to an equation.

Verification of Solutions

Verification of solutions is an essential last step in solving any equation to confirm that the solution is correct. It involves substituting the solution back into the original equation to check for accuracy. In this problem, we found that \( r = 10 \).

• To verify, substitute \( r = 10 \) back into the original form: \((5 \times 10 + 14)^{1/3} = 4\).
• Simplifying inside the parentheses, \( (50 + 14)^{1/3} = 4\).
• Thus, \( 64^{1/3} = 4 \), which confirms our earlier results.

Verification is crucial as it eliminates errors and ensures that the obtained solution is truly correct. It not only provides confidence in the accuracy of the result but also strengthens the understanding of the problem-solving process. Always remember to check your work, as practice makes perfect in mastering such techniques.