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Chapter 6: Problem 90

Find each root. See Section \(1.3 .\) $$ \sqrt[3]{27} $$

Short Answer

Expert verified
The cube root of 27 is 3.

Step by step solution

01

Understand the problem

Identify that the problem requires finding the cube root of 27, which is written as \(\sqrt[3]{27}\). The task is to determine which number, when multiplied by itself three times, gives 27.
02

Evaluate simple cube roots

Recall the basic cube numbers: \(1^3 = 1\), \(2^3 = 8\), and \(3^3 = 27\). Here, observe that \(3^3 = 27\), which implies that 3 is the number we are looking for.
03

Verify your result

Multiply the number 3 by itself three times: \(3 \times 3 \times 3 = 27\). Since the product is indeed 27, the cube root of 27 is correct.

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Key Concepts

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

Basic Cube Numbers
Cube numbers are the products of numbers multiplied by themselves twice. This means that the number is used as a factor three times. For example, if you take the number 2 and want to find its cube, you multiply it as follows:
  • First multiplication: \( 2 \times 2 = 4 \)
  • Second multiplication: \( 4 \times 2 = 8 \)
This gives you the result that \( 2^3 = 8 \). Cube numbers are important because they help in understanding cube roots. Here are a few basic cube numbers to be familiar with:
  • \( 1^3 = 1 \)
  • \( 2^3 = 8 \)
  • \( 3^3 = 27 \)
  • \( 4^3 = 64 \)
Recognizing these basic cube numbers helps us quickly evaluate and solve cube root problems.
Evaluating Roots
To evaluate a cube root, you need to determine which number, multiplied by itself three times, will give you the original number inside the radical. This is known as 'finding the cube root'. In the exercise provided, we look at the cube root of 27: First, we think about our basic cube numbers and match 27 with the appropriate cube number. Remembering the basic calculation, \( 3^3 = 27 \), we find that the cube root of 27 is 3. This is because:
  • First multiplication: \( 3 \times 3 = 9 \)
  • Second multiplication: \( 9 \times 3 = 27 \)
When learning to evaluate roots, it's useful to memorize or quickly reference basic cube numbers. This makes evaluating cube roots straightforward and accurate.
Verifying Solutions
After finding what you believe to be the cube root, it's crucial to verify your solution to ensure accuracy. Verification involves a simple step: multiplying your proposed solution by itself three times to see if it matches the original number within the cube root. Let's apply this to the cube root of 27. We believe the cube root is 3, so we calculate:
  • First: \( 3 \times 3 = 9 \)
  • Second: \( 9 \times 3 = 27 \)
Since multiplying 3 by itself three times gives us 27, we can be confident that our solution is correct. Verifying solutions is an excellent habit in mathematics, as it ensures that each part of your answer is correct before moving on to new problems.

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Most popular questions from this chapter

Try performing the following division without changing the order of the terms. Describe why this makes the process more complicated. Then perform the division again after putting the terms in the dividend in descending order of exponents. $$ \frac{4 x^{2}-12 x-12+3 x^{3}}{x-2} $$

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