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Chapter 9: Problem 116

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

Short Answer

Expert verified
The expression simplifies to \(-5xy \sqrt[3]{x^2 y}\).

Step by step solution

01

Recognizing the cube root

The expression involves a cube root, which is denoted by \( \sqrt[3]{a} \). This means finding a value that, when raised to the third power, equals \( a \). Our task is to simplify \( \sqrt[3]{-125 x^5 y^4} \).
02

Simplifying the numerical component

Identify the cube of a number that equals \(-125\). Since \(-5^3 = -125\), we can say \( \sqrt[3]{-125} = -5 \).
03

Simplifying variables with cube root

For \( x^5 \), find the largest cube you can make. Since \( x^3 \) is a cube (as \( 3 \) is the largest exponent less than \( 5 \) that's a multiple of \( 3 \)), we write \( x^5 = x^3 \cdot x^2 \). Thus, \( \sqrt[3]{x^5} = x \cdot \sqrt[3]{x^2} \).
04

Applying cube root to second variable

For \( y^4 \), the largest cube is \( y^3 \). Therefore, \( y^4 = y^3 \cdot y \), and \( \sqrt[3]{y^4} = y \cdot \sqrt[3]{y} \).
05

Combine all simplified components

The complete simplified expression is \(-5x(y)\) times the cube roots of the leftover parts. So, the expression simplifies to \(-5xy \sqrt[3]{x^2 y} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponents and Their Role
Exponents are a shorthand way of expressing repeated multiplication of a number by itself. For instance, when you see something like \( x^5 \), this means \( x \) is multiplied by itself five times: \( x \times x \times x \times x \times x \). Exponents are essential when dealing with cube roots, especially in expressions involving variables.To simplify expressions under a cube root, knowing how to break down the exponents into manageable parts is crucial. If we look at \( x^5 \), the actions here are:
  • First, recognize the cube \( x^3 \), which is the largest cube factor in \( x^5 \).
  • Split it into \( x^3 \cdot x^2 \) to match the cube root formula.
This method helps in reducing complex expressions into simpler forms, which makes solving and understanding variable expressions easier.
Deciphering Variable Expressions
Variable expressions like \( x^5 \) and \( y^4 \) are used in algebra to represent numbers with unknown values. Here, variables are raised to specific powers, indicating how many times the variable is used as a factor.When simplifying expressions with cube roots, we must evaluate the exponent parts carefully:
  • For instance, \( x^5 \) can be split using \( x^3 \) since \( 3 \) is the largest power of \( x \) divisible by \( 3 \).
  • This forms two parts: one inside the cube root and one outside, simplifying calculations considerably.
The same logic applies to \( y^4 \), where you split it into \( y^3 \) and \( y \). Understanding these variable expressions helps to manage and reorganize mathematical formulas effectively.
Parsing Radicals and Cube Roots
Radicals, especially cube roots, introduce another layer of complexity in mathematics. A cube root \( \sqrt[3]{a} \) intends to find a number which, when cubed, results in \( a \). Interpreting these requires some basic steps:For example, with \( -125 \), you might ask, "What value cubed gives me \(-125\)?" Since \( (-5)^3 = -125 \), the cube root is \(-5\).
  • This principle applies to the variables in the expression, such as simplifying \( x^5 \) using the largest cube you can extract, forming \( x^3 \cdot x^2 \).
  • Thus, \( \sqrt[3]{x^5} \) is split into simpler components like \( x \cdot \sqrt[3]{x^2} \).
Cube roots help in breaking down expressions into their simplest forms, allowing for more straightforward manipulations and understanding of complex radical terms.

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Most popular questions from this chapter

Solve each equation. Express all answers to four decimal places. $$ \ln x=0.926 $$

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