Problem 8 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ 3 \sqrt[3]{x}+1=16 $$

Step-by-Step Solution

Answer

Question: Solve the equation \(3\sqrt[3]{x} + 1 = 16\) for the variable x. Answer: The solution to the given equation is \(x=125\).

Step 1: Subtract 1 from both sides of the equation

To isolate the term with the cubic root, we will start by subtracting 1 from both sides of the equation: $$ 3\sqrt[3]{x} + 1 - 1 = 16 - 1 \\ 3\sqrt[3]{x} = 15 $$

Step 2: Divide by 3 from both sides of the equation

Now we will get rid of the 3 in front of the cubic root by dividing both sides of the equation by 3: $$ \frac{3\sqrt[3]{x}}{3} = \frac{15}{3} \\ \sqrt[3]{x} = 5 $$

Step 3: Cube both sides of the equation

To eliminate the cubic root and obtain the value of x, we will cube both sides of the equation: $$ (\sqrt[3]{x})^3 = 5^3 \\ x = 125 $$ So, the solution to the given equation is \(x=125\).

Key Concepts

Equation Solving

Solving an equation involves finding the value of the variable that makes the equation true. In the equation given, our goal is to find the value of \( x \) that satisfies the relation \( 3 \sqrt[3]{x} + 1 = 16 \). To do this, we perform a series of algebraic operations to isolate \( x \) on one side of the equation. Here is a brief rundown of the steps involved:

• Start by isolating terms that involve the variable. This often involves basic operations such as addition, subtraction, multiplication, or division. In our case, we subtract 1 from both sides to commence the isolation process.
• Next, handle any coefficients (numbers multiplied by the variable). This involves division if the coefficient is multiplying the term with the variable. In this exercise, we divide both sides by 3 to simplify the expression.
• Finally, resolve any roots or powers by taking the appropriate inverse operation. Here, we cube both sides to clear the cube root, revealing the value of \( x \).

These steps reflect common techniques used in equation solving, particularly with powers and roots, which are crucial in simplifying the expressions.

Cube Roots

Understanding cube roots is essential when dealing with cubic equations. A cube root of a number \( n \) is a value \( x \) such that \( x^3 = n \). In our equation \( 3 \sqrt[3]{x} = 15 \), the cube root symbol indicates we must find a value whose cube equals the original number under the root sign.Some important points about cube roots:

• Cube roots apply to both positive and negative numbers since both can be raised to odd powers (\( (-x)^3 = -x^3 \)).
• The cube root operation is the inverse of cubing a number; thus, if \( x^3 = 8 \), then \( \sqrt[3]{8} = 2 \).
• In our problem, determining the cube root was an intermediate step to isolate \( x \). After simplifying to \( \sqrt[3]{x} = 5 \), we eliminated the cube root by cubing both sides to solve for \( x \).

Understanding cube roots is crucial as it enables you to revert cubic representations back into their base numbers when solving related algebraic expressions.

Algebraic Manipulations

Algebraic manipulation involves strategically applying arithmetic operations to reshape an equation or algebraic expression. These manipulations are crucial when solving algebraic equations, like cubic ones, where such precision directly leads to the correct solution.Key manipulation strategies include:

• Balancing operations: Operations involving addition or subtraction must be applied equally to both sides to maintain equality. For instance, subtracting 1 from both sides in the initial step preserves the balance.
• Inverse operations: Applying inverse operations enables variable isolation. In our example, dividing by 3 undoes the multiplication of the cube root by 3, while cubing both sides removes the root.
• Simplifying expressions: Break down complex expressions into simpler forms. This helps us decode and simplify \( 3\sqrt[3]{x} = 15 \) to \( \sqrt[3]{x} = 5 \).

These techniques underline the problem-solving aspects of algebra, providing a toolkit to approach and solve various forms of equations efficiently and accurately. Being adept at manipulation paves the way for tackling even more complicated expressions confidently and with ease.