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Chapter 7: Problem 42

Find a simplified form of \(f(x) .\) Assume that \(x\) can be any real number. $$f(x)=\sqrt[3]{27 x^{5}}$$

Short Answer

Expert verified
The simplified form of \ f(x) = 3x \sqrt[3]{x^2}.

Step by step solution

01

Factor the expression under the cube root

The expression under the cube root is 27x^5, and it can be factored as 27*(x^3*x^2). This simplifies our process of taking the cube root.
02

Apply the cube root to each factor

Using the property \( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \), split the cube root: \( f(x) = \sqrt[3]{27} \cdot \sqrt[3]{x^3 \cdot x^2} \).
03

Simplify the cube roots

Simplify each cube root separately: \( \sqrt[3]{27} = 3 \) because 27 is 3 cubed, and \( \sqrt[3]{x^3 \cdot x^2} = x \cdot \sqrt[3]{x^2} \) because \( \sqrt[3]{x^3} = x \).
04

Combine the simplified components

Combine the simplified components to get \( f(x) = 3 \cdot x \cdot \sqrt[3]{x^2} \), which simplifies to \( f(x) = 3x \sqrt[3]{x^2} \).

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Key Concepts

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

Cube Roots
A cube root is a special value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).
The cube root symbol is represented as \( \sqrt[3]{...} \). It asks the question:
Properties of Exponents
Exponents play a crucial role in simplifying expressions. When dealing with roots, it鈥檚 essential to remember these key properties:
Factoring Under Roots
Factoring under roots is one of the most powerful tools to simplify radical expressions. Factoring is breaking down a number or expression into its prime numbers or simpler expressions that, when multiplied together, give the original number.

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