Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$(\sqrt[3]{-27})^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Evaluate the Cube Root

First, we need to find the cube root of \(-27\). The cube root of a number, \(-a\), is the value that, when cubed, results in \(-a\). \( -3^3 = -27 \), so \( \sqrt[3]{-27} = -3\).

Step 2: Square the Result from Step 1

Now that we have \( \sqrt[3]{-27} = -3 \), we need to square \(-3\) to find \((-3)^2\). \((-3)^2 = (-3) imes (-3) = 9\).

Key Concepts

Cube root

The concept of cube roots might seem challenging at first, but it's really about finding a number which, when multiplied by itself twice, equals the original number. In our example, we need to find the cube root of -27. The trick is to consider that we're looking for a number that satisfies \( x^3 = -27 \).For negative numbers like -27, take note that a negative number cubed always results in a negative number. Therefore, \(-3 \times -3 \times -3 = -27 \), making \( -3 \) the cube root of -27.

• The cube root of a positive number is positive.
• With a negative number, the cube root is naturally negative.

Whenever dealing with cube roots, it's beneficial to memorize cube values, such as \( 2^3 = 8 \) or \( 3^3 = 27 \), as this can make problem-solving faster.

Squaring numbers

Squaring a number means multiplying the number by itself. This is one of the simplest operations in mathematics but an important one. Consider the value -3 that was derived from our cube root.Squaring -3 might be confusing initially because you need to handle the negative sign correctly. Remember: multiplying two negative numbers together results in a positive number:

• \((-3) \times (-3) = 9\)

This is why \((-3)^2 = 9\).Practically:

• Positive numbers squared remain positive.
• Negative numbers squared become positive.

Paying attention to the signs is crucial when squaring; it ensures accurate results.

Negative numbers

Dealing with negative numbers often trips students up, but it doesn't have to. A negative number is simply a real number that's less than zero. It's worth remembering how negative numbers interact with mathematical operations like multiplication and division. When multiplying:

• Two negative numbers yield a positive result.
• A positive and a negative number result in a negative outcome.

When manipulating expressions involving negative numbers, watch for sign changes. This is especially important in operations like squaring and finding roots. To better handle negative numbers:

• Practice basic arithmetic operations with them.
• Use number lines to visualize.
• Pay careful attention to the context (e.g., subtractions often turn into additions).

Mastering negative numbers allows for fluent mathematical conversations and more accurate problem-solving.