Problem 97 - Algebra 2

Question

Simplify. Rationalize all denominators. $$6 \sqrt[3]{3}-2 \sqrt[3]{3}$$

Step-by-Step Solution

Answer

The simplified form is \(4 \sqrt[3]{3}\).

Step 1: Identify Like Terms

The cube root of 3 is the same in both \(6 \sqrt[3]{3}\) and \(2 \sqrt[3]{3}\). Therefore they are like terms and can be subtracted directly.

Step 2: Subtract Coefficients

Subtrack the coefficients of the like terms, \(6 - 2 = 4\).

Step 3: Present the Final Result

The final result is \(4 \sqrt[3]{3}\).

Key Concepts

Like Terms

When simplifying expressions, it's crucial to identify "like terms". "Like terms" are terms in an expression that have the same variable and the same exponent, which in this case, means their radical parts are identical. For instance, in the expression \(6 \sqrt[3]{3} - 2 \sqrt[3]{3}\), the cube root \(\sqrt[3]{3}\) is the same for both terms.

Recognizing like terms allows you to combine them, much like how you would with simple numbers or variables. You only need to operate on the coefficients, which are the numerical parts in front of the like terms. Here:

• The coefficients are 6 and 2.
• You subtract these coefficients directly.

By understanding that \(\sqrt[3]{3}\) is shared, you then perform the arithmetic on the coefficients to simplify the whole expression.

Rationalization

Rationalization is a process in mathematics used to eliminate the radicals from the denominator of a fraction. While this problem doesn't explicitly require rationalization as there is no denominator, understanding it can be very helpful.

The idea is straightforward: ensure any square, cube, or other roots found in a denominator are removed, creating a "rational" number without radicals in the denominator. Here's how it often works:

• Multiply the numerator and the denominator by the radical found in the denominator.
• This cancels the root in the denominator, creating a new fraction.

For example, if faced with \(\frac{1}{\sqrt{3}}\), multiply by \(\frac{\sqrt{3}}{\sqrt{3}}\) resulting in \(\frac{\sqrt{3}}{3}\). Thus, the final expression becomes free of radicals in the denominator.

Cube Roots

Cube roots are an interesting and frequently encountered mathematical concept, especially in algebra. The cube root of a number is what, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). It's denoted by the symbol \(\sqrt[3]{x}\).

Cube roots simplify similarly to how square roots do, except here you're looking at groups of three identical factors rather than two. When you see expressions like the ones in the problem \(6 \sqrt[3]{3}\), the \(\sqrt[3]{3}\) is the cube root of 3.

• If you have more complex numbers, you might need to calculate the exact cube root.
• For basic simplification, just manage the coefficients.

Understanding cube roots helps handle expressions with term complexities, allowing you to see potential simplifications and set up equations or inequalities correctly.