Problem 3 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 64\).

Step 1: Understand the Problem

We are given the equation \(x^{\frac{1}{3}} = 4\). This means we need to find a value for \(x\) such that when it is raised to the power of \(\frac{1}{3}\), the result is 4. In other words, find the cube of 4.

Step 2: Isolate x by Cubing Both Sides

To solve for \(x\), we need to eliminate the cube root. We do this by cubing both sides of the equation: \[(x^{\frac{1}{3}})^3 = 4^3\]Simplifying gives:\[x = 64\]

Step 3: Check the Solution

We can verify our solution by substituting \(x = 64\) back into the original equation to ensure it satisfies the equation. Calculate the cube root of 64: \[64^{\frac{1}{3}} = 4\]Since both sides of the equation equal 4, our solution is confirmed as correct.

Key Concepts

Understanding the Cube Root

When solving radical equations, it's vital to grasp the concept of cube roots. A cube root is a type of radical that tells us which number, when multiplied by itself three times (or cubed), results in the given number. For example, the cube root of 64 is 4, because \(4 \times 4 \times 4 = 64\). This notation is often symbolized as \(x^{\frac{1}{3}}\), which denotes the cube root of \(x\).
Understanding cube roots is essential because they help us simplify or solve equations involving radicals, especially in algebra where you might encounter \(x^{\frac{1}{3}} = 4\) or similar expressions.
To solve equations involving cube roots, one strategy is to raise both sides of the equation to the power of three, as this eliminates the radical and simplifies the expression, allowing you to isolate and solve for the variable.

The Process of Cubing

'Cubing' refers to raising a number to the power of 3. This operation is the inverse of finding a cube root. When you cube a number, you multiply it by itself twice: \(x^3 = x \cdot x \cdot x\). This process is useful in solving radical equations where isolating the variable involves removing cube roots.

In our example, when faced with \(x^{\frac{1}{3}} = 4\), cubing both sides of the equation is the key step. You execute this as follows:

• Cubing \(x^{\frac{1}{3}}\) gets you \(x^{\frac{1}{3} \times 3} = x\).
• Cubing 4 results in \(4^3 = 64\).

Thus, the equation simplifies to \(x = 64\). By determining the cube of numbers, we effectively eliminate radicals and solve for variables in such equations.

Checking Solutions in Radical Equations

Verifying your solution in radical equations is a crucial step to ensure accuracy. After solving equations like \(x^{\frac{1}{3}} = 4\), it's essential to substitute your found value back into the original equation to confirm its correctness.

For this problem, we found \(x = 64\). Plugging this back into the equation \(x^{\frac{1}{3}} = 4\), we compute:

• Calculate the cube root of 64: \(64^{\frac{1}{3}} = 4\).

The two sides of the equation match, confirming that the solution is correct. This final step of checking prevents potential errors and verifies that the solution satisfies the original problem, offering confidence in the answer's validity.