Problem 30 - Algebra A Combined Function

Question

Solve. $$ \sqrt[4]{2 x-9}-3=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 45\).

Step 1: Isolate the Radical Expression

The given equation is \( \sqrt[4]{2x - 9} - 3 = 0 \). To isolate the radical, add 3 to both sides to get:\[\sqrt[4]{2x - 9} = 3\]

Step 2: Eliminate the Fourth Root

To eliminate the fourth root, raise both sides of the equation to the power of 4:\[(\sqrt[4]{2x - 9})^4 = 3^4\]This simplifies to:\[2x - 9 = 81\]

Step 3: Solve for x

Now, solve the equation \(2x - 9 = 81\) for \(x\). Add 9 to both sides:\[2x = 90\]Then divide both sides by 2:\[x = 45\]

Step 4: Verify the Solution

Substitute \(x = 45\) back into the original equation to verify:\[\sqrt[4]{2(45) - 9} - 3 = \sqrt[4]{90 - 9} - 3 = \sqrt[4]{81} - 3\]Since \(\sqrt[4]{81} = 3\), we have:\[3 - 3 = 0\]Thus, the solution \(x = 45\) satisfies the original equation.

Key Concepts

Understanding Exponents and Radical Expressions

Exponents are a fundamental part of algebra. They signify how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. Exponents can greatly simplify expressions and calculations.

Radicals are the inverse operation of exponents. They involve finding roots and can be tricky to manipulate. The expression \(\sqrt[n]{x}\) refers to the \(n\)-th root of \(x\). In the given problem, \(\sqrt[4]{2x - 9}\), we deal with a fourth root, which means we're looking for a number that, when raised to the fourth power, equals \(2x - 9\).

Knowing how to handle these expressions is crucial when simplifying complex algebraic equations or solving problems involving polynomial roots.

Solving Radical Equations

Solving radical equations involves isolating the radical on one side and then eliminating it using the appropriate numerical power. This process generally requires several clear steps to ensure the solution is correct. Let's review how this works through our example.

• Start by isolating the radical expression. For the problem \(\sqrt[4]{2x - 9} - 3 = 0\), this involves adding 3 to both sides, giving \(\sqrt[4]{2x - 9} = 3\).
• Next, eliminate the root by raising both sides to the power of four: \((\sqrt[4]{2x - 9})^4 = 3^4\), resulting in \(2x - 9 = 81\). This is because raising a root to its corresponding power cancels out the radical, simplifying the equation.
• Finally, solve for the variable. Here, add 9 to both sides to obtain \(2x = 90\), and then divide by 2 to find \(x = 45\).

Each step methodically tackles the problem, helping to break down potentially overwhelming tasks into manageable parts.

Verification of Solutions

Verification of solutions is a crucial step in problem-solving. This ensures that no errors occurred during manipulation and that the solution indeed satisfies the original equation. To verify, we substitute our solution back into the original equation.

• Start by substituting \(x = 45\) back into \(\sqrt[4]{2x - 9} - 3 = 0\), which yields \(\sqrt[4]{90 - 9} - 3 = \sqrt[4]{81} - 3\).
• In this case, \(\sqrt[4]{81} = 3\), so we have \(3 - 3 = 0\). The left-hand side equals the right-hand side, which confirms that our solution is correct.

Verifying ensures that our solution is valid and not a product of calculation error. It builds confidence and a deeper understanding of the process used to achieve the conclusion.