Problem 116 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables represent positive numbers. $$\sqrt[3]{-125 x^{5} y^{4}}$$

Step-by-Step Solution

Answer

The simplified expression is \(-5xy\sqrt[3]{x^2y}\).

Step 1: Identify the Cube Root Expression

The expression given is \( \sqrt[3]{-125 x^{5} y^{4}} \). Our goal is to simplify this expression by finding the cube root of each term separately.

Step 2: Simplify the Numerical Term

First, break down \(-125\) as a cube: \(-125 = (-5)^3\). Therefore, the cube root of \(-125\) is \(-5\).

Step 3: Simplify the Variable \(x^5\)

The term \(x^5\) can be separated into \((x^3)\times(x^2)\). The cube root of \(x^3\) is \(x\), but \(x^2\) has no whole cube root and remains under the radical as \(\sqrt[3]{x^2}\).

Step 4: Simplify the Variable \(y^4\)

Similarly, the term \(y^4\) can be separated into \((y^3)\times(y)\). The cube root of \(y^3\) is \(y\), but \(y\) has no whole cube root and remains under the radical as \(\sqrt[3]{y}\).

Step 5: Combine the Terms

Combine all the simplified terms: The cube root of the expression is \(-5xy\sqrt[3]{x^2y}\), where \(-5xy\) are the simplified terms outside the radical and \(\sqrt[3]{x^2y}\) is the remaining expression under the cube root.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic statements into simpler, more manageable forms. The goal is to express the original expression in its simplest form, making calculations and understanding easier.

• Identify components: Look at each part of the expression separately, such as numbers, variables, and powers.
• Apply known rules: Use rules for operations like addition, subtraction, multiplication, and division of algebraic terms.
• Use factorization: This involves breaking down expressions into factors, which can make finding roots easier.

In our example, we simplify the expression \( \sqrt[3]{-125 x^{5} y^{4}} \) by taking cube roots of the components separately. By doing this, calculations become more straightforward and errors can be minimized.

Variables in Algebra

Variables are symbols that represent unknown or changeable values in algebraic expressions and equations. They are usually represented by letters like \(x\), \(y\), or \(z\).
Variables allow mathematicians to generalize mathematical problems and find solutions that apply to a range of numbers.
They are crucial in forming equations and expressions.

• Variables can be constants or symbolic representations of numbers that vary.
• They are often used to express relationships between different quantities.
• In our given problem, \(x\) and \(y\) are the variables we are working with.

Understanding how to work with them is essential when simplifying expressions, especially when roots and powers are involved.

Radicals in Algebra

Radicals are symbols used to indicate roots of quantities. The radical expression includes the radical sign (√) and the radicand, the number or expression inside the root.
Specifically, cube roots like \(\sqrt[3]{x}\) are used when we want to find a number that, when multiplied by itself three times, gives the original number.

• You can simplify radicals by factoring the number under the radical into its prime factors.
• Look for factors that are perfect cubes when dealing with cube roots, like \((-5)^3\) from -125.
• Sometimes, parts of the radicand do not result in whole roots and remain under the radical.

In the provided exercise, simplifying involves finding whole cube roots and separating the rest, like expressing \(y^4\) as \((y^3) \times (y)\) and taking \(\sqrt[3]{y^3}\) as \(y\), leaving \(\sqrt[3]{y}\) in the final expression. This step-by-step simplification makes working with radicals less daunting.