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Chapter 8: Problem 105

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

Short Answer

Expert verified
The simplified form of \( \sqrt[3]{\frac{27}{8}} \) is \( \frac{3}{2} \).

Step by step solution

01

Identify the Cube Roots

The first step is to identify the numbers under the cube root. Here, the numbers are 27 and 8. The cube root of 27 is 3, because 3 * 3 * 3 equals 27. Similarly, the cube root of 8 is 2, because 2 * 2 * 2 equals 8.
02

Apply the Cube Root

The next step is to apply the cube root to both numbers. The cube root of 27 is 3 and the cube root of 8 is 2. Therefore, the cube root of 27/8 is \( \frac{3}{2} \).
03

Write Down the Simplified Expression

The final step is to write down the simplified expression. The cube root of 27/8 simplifies down to \( \frac{3}{2} \).

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Key Concepts

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

Understanding Cube Roots
Let's dive into the concept of cube roots. The cube root of a number is simply another number that, when multiplied by itself three times, gives the original number. It's like asking, "What number can you cube (multiply three times by itself) to get this number?"
  • For instance, the cube root of 27 is 3 because: \[3 \times 3 \times 3 = 27\]
  • Similarly, the cube root of 8 is 2 because: \[2 \times 2 \times 2 = 8\]
Understanding this helps when you want to simplify expressions involving cube roots. It is crucial to recognize the perfect cubes like 8, 27, 64, etc., as they make your task easier. Knowing these can save time and reduce errors when you're solving problems.
Exploring Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. When dealing with radicals, you often see a symbol \( \sqrt[n]{\ldots} \), representing the root. The number outside the radical denotes the type of root:
  • The little number on the radical, like 2 or 3, indicates the degree of the root.
  • For square roots, it's commonly written without a number, as the square root is the default.
  • For cube roots, the number 3 indicates that it鈥檚 a cube root.
  • More complex roots could be represented by other numbers, such as 4 for fourth roots, and so on.
In our example, the cube root of \(\frac{27}{8}\) is written as \(\sqrt[3]{\frac{27}{8}}\). Radical expressions require simplification by factoring out perfect cubes or squares when applicable.
Simplification of Expressions
Simplifying expressions means rewriting them in their most compact and simplest form. This often involves removing radicals by finding their root, factoring, or canceling like terms.Steps to simplify the cube root of \(\frac{27}{8}\):
  • Identify the perfect cube factors. In \(\frac{27}{8}\), both 27 and 8 are perfect cubes.
  • Apply the cube root to both the numerator and the denominator individually, such as: \[\sqrt[3]{27} = 3\ \,\, \text{and}\ \,\, \sqrt[3]{8} = 2\]
  • Write the simplified form as \(\frac{3}{2}\).
It's important to simplify your answer so it's easier to interpret and compare with other expressions. Mastering simplification can make complicated expressions more manageable and clearer.

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Most popular questions from this chapter

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

In Exercises \(94-97,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 \sqrt{x}}{x \sqrt{6}}=\frac{\sqrt{6 x}}{2 x} \text { for } x>0$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$32^{-\frac{4}{5}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$16^{-\frac{3}{4}}$$

$$\text { Simplify: } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}$$
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