Problem 74 - Intermediate Algebra

Question

Simplify each cube root. See Example 6. $$ \sqrt[3]{-27} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{-27}\) is \(-3\).

Step 1: Identify the Cube Root Problem

The cube root expression given is \(\sqrt[3]{-27}\). A cube root asks us to find a number that, when multiplied by itself three times, gives the original number \(-27\).

Step 2: Determine the Cube Root of Positive 27

First, determine the cube root of \(27\). We know that \(3 \times 3 \times 3 = 27\). Thus, the cube root of \(27\) is \(3\), or \(\sqrt[3]{27} = 3\).

Step 3: Apply Cube Root to Negative Number

Since we are dealing with \(-27\), recall that the cube of a negative number is also negative. Therefore, \(-3 \times -3 \times -3 = -27\). Hence, the cube root of \(-27\) is \(-3\).

Step 4: Write Final Simplified Form

Conclude by stating the simplified form of the original expression: \(\sqrt[3]{-27} = -3\).

Key Concepts

Understanding Cube Roots

When dealing with cube roots, it's essential to understand what they represent. Simply put, a cube root of a number is a value that, when multiplied by itself three times, will result in the original number. For example, consider the number 8. The cube root of 8 is 2 because when you multiply 2 three times, you get 8:

• \[2 \times 2 \times 2 = 8\]

The notation for cube root is different from a square root, as it includes the index number 3, indicating that the root is of the third degree. This is expressed as \(\sqrt[3]{x}\). When simplifying cube roots, it's important to identify the factor that, when cubed, will return the original number. This process applies to both positive and negative numbers.

Working with Negative Numbers

Cube roots can apply to both positive and negative numbers. This is because multiplying a negative number by itself three times still results in a negative number, due to the properties of arithmetic operations.
For instance, let's explore the cube root of \(\ -27 \). When looking for what multiplied by itself three times gives \(\ -27 \), we consider \(\ -3 \):

• \[-3 \times -3 \times -3 = -27\]

This shows us that the cube root of a negative number remains negative.
Thus, the concept of cube roots extends neatly to negative numbers without requiring any complex numbers. This highlights an important difference from square roots, which do not yield real results for negative numbers because squares are always positive.

Simplifying Radicals Further

Simplifying radicals, especially cube roots, involves identifying the prime factors or perfect cubes within a number. By doing so, we can find more straightforward representations of these mathematical expressions.
The key is to determine whether a number is a perfect cube or can be broken into smaller perfect cubes. If a number like 64 can be expressed as a cube, i.e., \((4 \times 4 \times 4)\), it simplifies directly to 4 without further need for estimation or calculation.

• Perfect cubes include numbers like 1 (since \(1 \times 1 \times 1\)), 8 (\(2 \times 2 \times 2\)), 27 (\(3 \times 3 \times 3\)), and so forth.
• To factor a larger cube, identify smaller perfect cube components within it that can be broken down into simpler terms.

Breaking down radicals in this way not only makes expressing and handling them easier but also aids in quick mental calculations. This is a handy tactic to simplify complex problems into more manageable equations.