Problem 6 - Algebra 2

Question

Solve each equation. $$ \sqrt[3]{x-4}=3 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 31 \).

Step 1: Understand the Equation

The equation given is \( \sqrt[3]{x-4} = 3 \). Here, \( \sqrt[3]{...} \) denotes the cube root of the expression inside the parentheses. Our task is to solve for \( x \).

Step 2: Eliminate the Cube Root

To remove the cube root, we need to cube both sides of the equation. Doing this, we have:\[(\sqrt[3]{x-4})^3 = 3^3\]This simplifies to:\[x - 4 = 27\]

Step 3: Solve for x

Now, we have a simple equation, \( x - 4 = 27 \). To solve for \( x \), add 4 to both sides:\[x = 27 + 4\]\[x = 31\]

Step 4: Verify the Solution

To ensure that our solution is correct, substitute \( x = 31 \) back into the original equation:\[\sqrt[3]{31-4} = \sqrt[3]{27} = 3\]Since both sides of the equation are equal, our solution is verified.

Key Concepts

Cube Root

In mathematics, the cube root of a number is a special value that, when multiplied by itself twice, gives the original number. In simpler terms, if you have a number under the cube root symbol, finding the cube root means figuring out which number multiplied three times results in the given number. For example, the cube root of 27 is 3 because multiplying 3 by itself twice (3 \(\times\) 3 \(\times\) 3) equals 27.
Cube roots play a vital role in solving equations that involve expressions under the cube root symbol. In our problem, the expression \( \sqrt[3]{x-4} = 3 \) means finding the value of \( x \) that makes this equation true.

Equation Verification

Equation verification is an essential step in solving algebraic equations because it confirms that the solution is correct. Verification involves substituting the value obtained back into the original equation and checking if the equality holds true.
In our example, after calculating \( x = 31 \), we substitute \( 31 \) back into the original equation \( \sqrt[3]{x - 4} = 3 \). We perform the operation \( \sqrt[3]{31 - 4} = \sqrt[3]{27} \) which simplifies to 3, verifying our solution. This method ensures that no mistakes were made during the process of solving the equation.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic operations to simplify and solve equations. This involves steps like combining like terms, adding or subtracting terms from both sides, and eliminating complex terms like roots or exponents.
In the problem \( \sqrt[3]{x-4} = 3 \), the first step in algebraic manipulation is to eliminate the cube root by cubing both sides. This simplifies the equation to \( x - 4 = 27 \).

• Next, simple addition is used to solve for \( x \), by adding 4 to both sides, resulting in \( x = 31 \).
• Such techniques help in breaking down complex problems into smaller, manageable steps, making it simpler to arrive at a solution.

Mastering algebraic manipulation is crucial, as it is widely used in various types of equations and mathematical problems.