Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ 2+\sqrt[3]{1-5 t}=0 $$

Step-by-Step Solution

Answer

The solution is \( t = \frac{9}{5} \).

Step 1: Isolate the Cubic Root

Begin by isolating the cubic root expression on one side of the equation. Subtract 2 from both sides to do this: \[ \sqrt[3]{1-5t} = -2 \]

Step 2: Eliminate the Cubic Root

Eliminate the cubic root by raising both sides of the equation to the power of 3: \[ (\sqrt[3]{1-5t})^3 = (-2)^3 \]This simplifies to: \[ 1-5t = -8 \]

Step 3: Solve for t

Now, solve for \( t \). First, subtract 1 from both sides: \[ -5t = -8 - 1 \]Simplify the right side to get: \[ -5t = -9 \]Then divide both sides by -5 to isolate \( t \): \[ t = \frac{-9}{-5} = \frac{9}{5} \]

Key Concepts

Cubic Roots

Cubic roots are not as commonly encountered as square roots but are an important part of algebraic expressions. A cubic root of a number is a value that, when cubed (multiplied by itself twice), gives the original number. The notation used to indicate a cubic root is \( \sqrt[3]{x} \), where **3** is the index indicating it is a cubic root.
Understanding cubic roots is crucial in solving equations like \( 2 + \sqrt[3]{1-5t} = 0 \). Here, the term \( \sqrt[3]{1-5t} \) represents a cubic root. When you see it, the goal often is to eliminate the root to simplify the equation. You can do this by raising both sides of the equation to the power of 3, which reverses the effect of the cubic root. This turns the expression \( \sqrt[3]{1-5t} \) into \( 1-5t \), eliminating the complexity of dealing with roots.

Isolating Variables

Isolating a variable is key to simplifying and solving equations. This means rearranging the equation, so the variable you're solving for is alone on one side. For the equation \( 2 + \sqrt[3]{1-5t} = 0 \), our first goal is to isolate the cubic root term.
This is done by moving other terms to the opposite side of the equation. In the given exercise, subtracting 2 from both sides isolates the cubic root: \( \sqrt[3]{1-5t} = -2 \). By performing this operation, you focus on the main expression involving the variable, simplifying later steps.
Remember, the operations applied to both sides of the equation must maintain the balance of the equation. Isolating variables step-by-step might seem slow, but it's a powerful tool that allows you to solve equations systematically.

Solving Equations

Solving equations involves multiple steps: isolating variables, removing roots or exponents, and simplifying expressions to find the value of unknowns. For the equation \( \sqrt[3]{1-5t} = -2 \), after isolating the variable's root, you eliminate the cubic root by raising both sides to the third power:

• Expanding \( (-2)^3 \) results in \(-8\), transforming the equation to \( 1-5t = -8 \).
• Next, you simplify further to isolate \( t \). This includes subtracting 1 from both sides, resulting in \(-5t = -9\).
• Lastly, division by -5 gives \( t = \frac{9}{5} \), completing the solution.

This systematic approach ensures that each step logically follows the last, maintaining the integrity of the equation. Solving equations can initially seem challenging, but meticulous step-by-step techniques simplify this task greatly.