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Chapter 10: Problem 20

Find each cube root. $$\sqrt[3]{27}$$

Short Answer

Expert verified
The cube root of 27 is 3.

Step by step solution

01

Understand the Problem

We need to find the cube root of 27, which means we need to find a number that, when multiplied by itself three times, equals 27.
02

Identify the Perfect Cube

Recognize that 27 is a perfect cube. This means there is an integer which, when cubed, gives 27.
03

Find the Cube Root

Start testing integers by cubing them to see which one gives us 27. We find that \(3^3 = 3 \times 3 \times 3 = 27\).
04

Verify the Solution

Verify that multiplying 3 by itself twice more indeed results in 27 to confirm that 3 is the cube root of 27.

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Key Concepts

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

Perfect Cubes
Understanding perfect cubes is crucial when dealing with cube roots. A perfect cube is a number that can be expressed as the product of an integer multiplied by itself three times. In simpler terms, if you take any whole number and multiply it by itself, then multiply again, you will get a perfect cube. For example, if you take the number 3 and do the math:
  • First multiply: 3 x 3 = 9
  • Second multiply: 9 x 3 = 27
Thus, 27 is a perfect cube, with 3 being the integer that gives us 27 when cubed. Recognizing perfect cubes helps us easily find cube roots without having to use calculators or complex operations.
Integer Solutions
Cube roots are easier to manage when the number under the cube root is a perfect cube. Perfect cubes have integer solutions. An integer solution is simply a whole number solution without any decimal or fractional parts. For example, the cube root of 27 involves finding what integer multiplied by itself three times gives us 27.

By testing a few numbers, we find that 3 fits perfectly because:
  • 3 x 3 x 3 = 27
There are no fractions or decimals here; we strictly work with whole numbers. Having integer solutions simplifies calculations and makes comprehension significantly easier, especially in beginner math problems.
Mathematical Verification
Verification in math is about confirming your solution is correct. It's a way to double-check your work and ensure accuracy. After finding that 3 is the cube root of 27, a thorough verification involves retracing the steps to ensure no mistake was made.

For instance:
  • Calculate: 3 x 3 = 9
  • Then: 9 x 3 = 27
This confirms that indeed 3 multiplied by itself three times equals the original number 27. Verification is essential, especially when solving math problems by hand. It helps catch errors and deepen your understanding of the problem-solving process.

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Most popular questions from this chapter

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