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Chapter 5: Problem 5

Find the indicated real \(n\)th \(\operatorname{root}(\mathrm{s})\) of a. \(n=3, a=8\)

Short Answer

Expert verified
The cube root of 8 is 2.

Step by step solution

01

Understand the \(n\)th root

The \(n\)th root of a number \(a\) is a number \(x\) such that when \(x\) is raised to the power of \(n\), it equals \(a\). When \(n = 3\), we are looking for the cube root, which means that we need to find a value of \(x\) that when cubed (or raised to the power of 3) gives 8.
02

Calculate the \(n\)th root

To solve the cube root of 8 (鈭�8), look for a number that when multiplied by itself twice (or cubed) equates to 8. By trial and error, or knowing the basic cubes of natural numbers, it is known that \(2^3 = 8\). Hence, the cube root of 8 is 2.

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Key Concepts

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

Cube Root
The cube root of a number is an essential concept when understanding powers and radicals. A cube root is the value that, when taken three times in multiplication, results in the original number. For instance, the cube root of 8 is 2 because \(2 imes 2 imes 2 = 8\). The cube root is often represented with the radical symbol and a small three, like this: \( \sqrt[3]{a} \). This is different from square roots, which look similar but have a small two or no number at all.
  • A cube root essentially "undoes" a cube operation, returning the base number if you know one of its cube powers.
  • Understanding cube roots helps in distinguishing them from other roots, where the index would change based on the power, like fourth roots or square roots.
Cube roots are particularly useful when dealing with geometry, especially in calculating dimensions like volume, as those often involve three dimensions.
Exponents
Exponents play a crucial role in simplifying the representation of repeated multiplication. When a number is raised to the third power, like \(x^3\), it means \(x imes x imes x\). Here, 3 is the exponent, or power, indicating how many times the base, x, is used in a multiplication. This concept is tied closely to the notion of roots, including cube roots.
  • Exponents allow concise expression of large numbers or operations, reducing manual calculation errors.
  • They provide a foundation for learning more advanced math topics, such as polynomials and exponential functions.
When you encounter exponents in problems like finding the cube root, it often requires recognizing that the reverse of a power, like going from \(x^3\) to its cube root, is a fundamental shift in understanding mathematical operations.
Radicals
Radicals are symbols used to indicate roots of a number and are extensively used in algebra and higher mathematics. A radical sign (\(\sqrt{}\)) includes the number or expression underneath referred to as the radicand. For cube roots, an index of 3 is placed above the radical. In expressions, this ensures clarity regarding the type of root.
  • The radicand is the number you are trying to find the root of.
  • By knowing the radical's index, you can determine which power to apply for reversing the root operation.
Radicals help simplify complex expressions and solve equations, as they can untangle exponents to yield solutions in a more straightforward form. Understanding how to interpret and convert between radical forms and exponent notations is fundamental for problem-solving in mathematics.

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