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Chapter 8: Problem 105

Simplify each radical. $$ \sqrt[3]{\frac{8}{27}} $$

Short Answer

Expert verified
The simplified form is \(\frac{2}{3}\).

Step by step solution

01

- Identify the Cube Roots

Recognize that the cube root \(\frac{8}{27}\) can be simplified by taking the cube root of the numerator and the cube root of the denominator separately. Recall that \(a \cdot b = c \Rightarrow \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{c}\).
02

- Simplify the Numerator

Find the cube root of the numerator: \( \sqrt[3]{8} = 2 \). That's because 2 \times 2 \times 2 = 8.
03

- Simplify the Denominator

Find the cube root of the denominator: \( \sqrt[3]{27} = 3 \). That's because 3 \times 3 \times 3 = 27.
04

- Combine the Results

Combine the results from the numerator and the denominator: \(\frac{ \sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}\).

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Key Concepts

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

Radical Simplification
Radical simplification involves breaking down a radical expression to its simplest form. This process is especially important when working with cube roots. The expression is simplified by identifying and separating the radical operations for the numerator and the denominator. For instance, in the exercise \( \sqrt[3]{ \frac{8}{27} }\), we start by simplifying both the numerator and the denominator individually under the cube root. Recognizing that \( \sqrt[3]{ \frac{a}{b} }\) equals \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \), we can handle each part separately. This step-by-step method makes complex fractions manageably simple.
Numerator and Denominator
The terms 'numerator' and 'denominator' are fundamental in fraction operations. The numerator is the top part of the fraction, while the denominator is the bottom part. In the exercise, our fraction begins as \( \frac{8}{27} \) under a cube root. To simplify:
  • First, find the cube root of the numerator: \( \sqrt[3]{8} = 2 \). This is because 2 \times 2 \times 2 equals 8.
  • Next, find the cube root of the denominator: \( \sqrt[3]{27} = 3 \). This is because 3 \times 3 \times 3 equals 27.

After simplifying both parts, recombine them to get the simplified fraction \( \frac{2}{3} \). Recognizing and calculating cube roots individually for numerator and denominator helps manage more complex expressions.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Simplifying these requires a combination of algebraic rules and radical operations. For radicals like cube roots, the fraction remains manageable through careful step-by-step simplification. Look at our initial fraction \( \sqrt[3]{ \frac{8}{27} }\). Initially complex, simplifying the cube root of the numerator and denominator separately shows:
  • Numerator simplifies to 2
  • Denominator simplifies to 3

So, \( \sqrt[3]{ \frac{8}{27} } \) simplifies to \( \frac{2}{3} \). By breaking the process into smaller steps, working with algebraic fractions with radical components becomes more straightforward and less intimidating.

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Most popular questions from this chapter

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{25 t^{11}} $$

Simplify each radical. $$ \sqrt[3]{y^{18}} $$

Find each product and simplify. Simplify the product \(\sqrt{8} \cdot \sqrt{32}\) in two ways. First, multiply 8 by 32 and simplify the square root of this product. Second, simplify \(\sqrt{8},\) simplify \(\sqrt{32, \text { and then multiply. }}\) How do the answers compare? Make a conjecture (an educated guess) about whether the correct answer can always be obtained using either method when simplifying a product such as this.

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{900 y^{8}} $$

Simplify each radical. $$ \sqrt[3]{w^{3}} $$
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