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Chapter 0: Problem 79

Add or subtract terms whenever possible. $$\sqrt[3]{54 x y^{3}}-y \sqrt[3]{128 x}$$

Short Answer

Expert verified
The simplified expression is \((3x-y) * 4x\sqrt[3]{2}\)

Step by step solution

01

Break down the cube roots

First step involves breaking down the cube roots into simpler terms. The cube root of 54 can be broken down as \(\sqrt[3]{27*2}\) or \(3\sqrt[3]{2}\), similarly, the cube root of 128 can be broken down as \(\sqrt[3]{64*2}\) or \(4\sqrt[3]{2}\), and finally, \(y^3\) as \(y*y*y\). This makes the expression as \(3x\sqrt[3]{2}y-y*4x\sqrt[3]{2}\).
02

Combine like terms

Now we can observe that both terms share a common factor of \(x\sqrt[3]{2}y\). We can combine these like terms to simplify the expression further, leading to \((3x-y) * 4x\sqrt[3]{2}\)

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Key Concepts

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

Cube Roots
Understanding cube roots is essential when simplifying expressions involving them. A cube root, denoted as \( \sqrt[3]{a} \), is a number that, when multiplied by itself twice, yields the original number \( a \). For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \). When simplifying expressions with cube roots like \( \sqrt[3]{54} \), breaking it down into its prime factors helps identify perfect cubes. In this case, \( 54 = 27 \times 2 \), so \( \sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3 \sqrt[3]{2} \). This breakdown helps isolate and simplify cube roots in expressions, making addition or subtraction of terms easier to handle.
Factoring Expressions
Factoring is a technique used in algebra to simplify expressions or solve equations by breaking down numbers or expressions into products of their factors. When we factor an expression, we look for common factors that can be extracted or simplified further. For example, in the expression \( \sqrt[3]{54xy^3} \), we can factor each component: \( 54 = 27 \times 2 \), and \( y^3 = y \times y \times y \). This results in the expression being factored into \( 3xy \sqrt[3]{2} \). Recognizing and extracting these factors often makes it easier to simplify the entire expression. Factoring is an invaluable skill in algebra, especially when dealing with polynomials or complex expressions.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions using a variety of mathematical operations. In the original exercise, both terms share a common mathematical expression, \( x \sqrt[3]{2} y \). By identifying these like terms, the expression becomes \( 3x \sqrt[3]{2} y - y(4x \sqrt[3]{2}) \).
  • Step 1: Reducing it, we can factor out the common term \( x \sqrt[3]{2} y \), leading to \( (3x - 4xy) \sqrt[3]{2} \).
  • Step 2: Further simplification like \( (3x - 4xy) \) shows us that these terms are simplified as much as possible given no further like terms.
Algebraic manipulation is crucial in simplifying expressions and making equations easier to manage, proving invaluable in problem-solving situations.

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