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Chapter 5: Problem 8

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

Short Answer

Expert verified
The expression evaluates to 9.

Step by step solution

01

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 \( \sqrt[3]{-27} = -3\).
02

Square the Result from Step 1

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

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Key Concepts

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

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.

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