Problem 26 - Algebra 2

Question

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt[3]{250 x^{7} y^{3}}}{\sqrt[3]{2 x^{2} y}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(5xy \sqrt[3]{\frac{x}{y}}\).

Step 1: Simplify the cube roots

We simplify each of the cube roots separately. For the numerator \(\sqrt[3]{250 x^{7} y^{3}}\), it can be simplified to \((5x^{2}y)\sqrt[3]{2x}\). For the denominator \(\sqrt[3]{2 x^{2} y}\), this simplifies to \(x\sqrt[3]{2y}\). Therefore, the expression simplifies to the fraction \(\frac{(5x^{2}y)\sqrt[3]{2x}}{x\sqrt[3]{2y}}\).

Step 2: Divide the Fractions

Next, divide the corresponding parts of the fraction. Divide \(5x^{2}y\) by \(x\), which simplifies to \(5xy\). Then divide \(\sqrt[3]{2x}\) by \(\sqrt[3]{2y}\). This simplifies to \(\sqrt[3]{\frac{x}{y}}\). This results in the expression \(5xy \sqrt[3]{\frac{x}{y}}\).

Step 3: Check for further simplification

The expression \(5xy \sqrt[3]{\frac{x}{y}}\) cannot be simplified further. There are no like terms or redundant terms to eliminate or simplify.

Key Concepts

Cube Roots

A cube root is a special type of root that undoes a cubing process. It answers the question: "What number multiplied by itself three times gives this number?" Mathematically, if you have a number or an expression such as 8 or \( x^3 \), the cube root finds what number, when cubed, results in that expression. The cube root of a number \( a \), written as \( \sqrt[3]{a} \), is essentially \( b \) such that \( b^3 = a \).
Understanding cube roots is essential when simplifying expressions that involve variables raised to powers that can be divided by three. In our original exercise, the cube roots, \( \sqrt[3]{250 x^{7} y^{3}} \) and \( \sqrt[3]{2 x^{2} y} \), must be simplified before proceeding to division, breaking complex expressions into simpler factors.
When breaking down the cube roots:

• Identify perfect cubes: These are numbers like 1, 8, 27, and expressions \( x^3, y^3 \).
• Extract cube root principles: For variables, extract by dividing exponents by 3.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while retaining the original value. This process may involve combining like terms, reducing fractions, and eliminating unnecessary complexity in operations or components, like with roots or exponents.
In math, a simplified expression is easier to understand and work with.
In the exercise provided, the expression \( \sqrt[3]{250 x^{7} y^{3}} \) is simplified step-by-step to \((5x^{2}y)\sqrt[3]{2x}\). Here is a breakdown:

• Separate constant from variables: Decompose 250 into \( 5^3 \times 2 \).
• Divide exponents by 3: For \( x^7 \) becomes \( x^2 \) with a remainder \( x \).
• Apply combined terms: Collect results like \( 5x^{2}y \) for easy evaluation.

Taking each factor, numeric and variable, and simplifying each separately helps gain more control over evaluating expressions, which is critical in algebra.

Algebraic Fractions

Algebraic fractions are similar to regular fractions but involve algebraic expressions in the numerator, the denominator, or both. The goal is to simplify them to make solving and analyzing more straightforward.
Simplifying algebraic fractions involves dividing out common factors and reducing expressions effectively.
The given problem displayed an algebraic fraction \( \frac{(5x^{2}y)\sqrt[3]{2x}}{x\sqrt[3]{2y}} \), and asked to simplify it:

• Identify common terms: Divide \( x \) in fractions to get \( 5xy \).
• Simplify cube root division: \( \frac{\sqrt[3]{2x}}{\sqrt[3]{2y}} \) reduces to \( \sqrt[3]{\frac{x}{y}} \).

By combining these results, \( 5xy \times \sqrt[3]{\frac{x}{y}} \) represents the fully simplified version. Understanding and simplifying algebraic fractions is vital to solving fractional equations and interpreting complex expressions. It also assists in further exploring algebraic concepts such as polynomials and rational expressions.