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Chapter 10: Problem 29

Simplify by factoring. $$\sqrt[3]{27 x^{3}}$$

Short Answer

Expert verified
The simplified form of \( \sqrt[3]{27 x^{3}} \) is 3x

Step by step solution

01

Identify the perfect cubes

In the given expression, 27 and \(x^{3}\) are perfect cubes. The cube root of 27 is 3 because \(3^{3} = 27\). The cube root of \(x^{3}\) is x because \(x^{3}\) is the cube of x.
02

Simplify the cube root

After identifying the cube roots, simplify the given expression. The cube root of 27 is 3 and the cube root of \(x^{3}\) is x. Therefore, \( \sqrt[3]{27 x^{3}} = 3x\).

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Key Concepts

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

Perfect Cubes
When we talk about perfect cubes in mathematics, we are referring to numbers or expressions that can be written as the cube of a whole number or variable. For example, 27 is a perfect cube because it equals \(3^3\) or 3 cubed. Additionally, an expression like \(x^3\) is also a perfect cube because it's the result of \(x\times x\times x\).
To identify a perfect cube:
  • Find a number that, when multiplied by itself three times, equals the given number or expression.
  • Check if the term can be expressed in the form \(a^3\), where \(a\) is a whole number or an algebraic expression.
Recognizing perfect cubes helps simplify expressions, especially when working with cube roots.
Cube Root Simplification
Cube root simplification is the process of breaking down a cube root expression into its simplest form. To simplify a cube root, you need to find the cube root of each part of the expression.
Consider \(\sqrt[3]{27x^3}\):
  • The cube root of 27 is 3, because \(3^3 = 27\).
  • The cube root of \(x^3\) is \(x\), as \(x^3\) equals \(x\times x\times x\).
Combining these, the cube root of \(27x^3\) simplifies to \(3x\). This simplification process involves extracting the cube root of each component of an expression to achieve a reduced form. Simplifying cube roots makes calculations easier and results more manageable.
Algebraic Expressions
Algebraic expressions are combinations of constants, variables, and operations (like addition, subtraction, multiplication, and division) that together form a mathematical phrase. Examples of algebraic expressions include \(3x^2 + 2x - 5\) and \(x^3 + 27\).
Here are key points about algebraic expressions:
  • They can represent numbers or quantities, often involving variables that allow for dynamic calculations.
  • Variables within expressions can be simplified or evaluated for specific values, providing insight into the relationships between numbers.
  • Simplifying algebraic expressions such as \(\sqrt[3]{27x^3}\) involves recognizing patterns, like perfect cubes, and applying rules like cube root extraction.
Understanding algebraic expressions and how to simplify them is crucial for solving equations and performing algebraic manipulations efficiently.

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Most popular questions from this chapter

Use a graphing utility to solve each radical equation. Graph each side of the equation in the given viewing rectangle. The equation's solution set is given by the \(x\) -coordinate(s) of the point (s) of intersection. Check by substitution. $$\begin{aligned} &\sqrt{x^{2}+3}=x+1\\\ &[-1,6,1] \text { by }[-1,6,1] \end{aligned}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{2 \sqrt{6}+\sqrt{5}}{3 \sqrt{6}-\sqrt{5}}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{\sqrt{11}-\sqrt{5}}{\sqrt{11}+\sqrt{5}}$$

Determine whether each relation is a function. (Section 8.1, Example 2) a. \(\\{(-1,1),(1,1),(-2,4),(2,4)\\}\) b. \([(1,-1),(1,1),(4,-2),(4,2)]\)

In Exercises \(39-64,\) rationalize each denominator. $$\frac{10}{\sqrt[5]{16 x^{2}}}$$
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