Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-17\) solve each equation and check. $$ 14-4 b^{\frac{1}{3}}=2 $$

Step-by-Step Solution

Answer

The solution is \( b = 27 \).

Step 1: Isolate the term with the variable

Start with the equation: \[ 14 - 4b^{\frac{1}{3}} = 2 \]Subtract 14 from both sides to isolate the term with \( b \):\[ -4b^{\frac{1}{3}} = 2 - 14 \]This simplifies to:\[ -4b^{\frac{1}{3}} = -12 \]

Step 2: Solve for the fractional exponent term

Divide both sides by \(-4\) to isolate \( b^{\frac{1}{3}} \):\[ b^{\frac{1}{3}} = \frac{-12}{-4} \]Simplifying gives:\[ b^{\frac{1}{3}} = 3 \]

Step 3: Remove the fractional exponent

To solve for \( b \), cube both sides to eliminate the fractional exponent:\[ (b^{\frac{1}{3}})^3 = 3^3 \]This results in:\[ b = 27 \]

Step 4: Check the solution

Substitute \( b = 27 \) back into the original equation to verify it works:\[ 14 - 4(27^{\frac{1}{3}}) = 2 \]Since \( 27^{\frac{1}{3}} = 3 \), substitute:\[ 14 - 4 \times 3 = 2 \]Calculate the left side:\[ 14 - 12 = 2 \]The equation holds true, confirming the solution is correct.

Key Concepts

Isolate the Variable

One of the most important skills in algebra is learning how to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. This helps us to find the solution to the equation.
In the given exercise, we start with the equation \(14 - 4b^{\frac{1}{3}} = 2\). Our first goal is to move all other terms to the opposite side by performing reverse operations.

• First, subtract 14 from both sides to get: \(-4b^{\frac{1}{3}} = 2 - 14\).
• This simplifies to \(-4b^{\frac{1}{3}} = -12\), helping isolate the term that includes the variable.

Perform operations step by step, and be sure to do the same operation on both sides.
Make sure that your aim is always to get the variable term alone on one side.

Fractional Exponents

A fractional exponent like \(b^{\frac{1}{3}}\) means that the base, \(b\), is being raised to the power of \(\frac{1}{3}\). Fractional exponents are another way to represent roots.
For \(b^{\frac{1}{3}}\), the exponent \(\frac{1}{3}\) indicates the cube root of \(b\). To eliminate the fractional exponent, we need to perform an operation that will bring it back to a full power.

• Cubing both sides of the equation \((b^{\frac{1}{3}} = 3)\) effectively removes the fractional exponent.
• We use \((b^{\frac{1}{3}})^3\) to get back to \(b\).
• This results in \(b = 3^3\), which simplifies to \(b = 27\).

Understanding fractional exponents is crucial as they frequently occur in equations involving roots and powers.
They help simplify complex problems.

Solving Equations Step-by-Step

Solving equations step-by-step involves breaking down the problem into manageable parts. Each step builds on the previous. Consistent careful work is important to find the correct solution.

• Step 1: Isolate the variable term, as done in the first section.
• Step 2: Solve for the part with the fractional exponent, focusing on simplifying fractions.
• Step 3: Remove or transform the fractional exponent by applying the inverse operation.
• Step 4: Verify your solution by substituting back into the original equation.

By verifying, we confirm the solution's accuracy.
The step-by-step method encourages thoroughness and helps prevent errors in calculations.
Practice makes these processes second nature, helping you confidently tackle any algebraic equation.