91Ó°ÊÓ

A perfect cube is a number that can be expressed as the cube of an integer. It is a result of multiplying an integer by itself two more times; for example, the cube of 2 is 2 × 2 × 2 = 8. In a broader sense, perfect cubes can be derived from both positive and negative integers. This unique characteristic makes them interesting because they include numbers like -8 or -27, which result from cubing negative integers such as -2 or -3 respectively.
Understanding perfect cubes is crucial for evaluating cube roots. They serve as the foundation for recognizing numbers that can be easily simplified using cube roots. The knowledge of perfect cubes enables students to convert complicated expressions into more manageable ones.
Here are some examples of perfect cubes:

• 1 (since 1³ = 1)
• 8 (since 2³ = 8)
• 27 (since 3³ = 27)
• -8 (since (-2)³ = -8)
• -125 (since (-5)³ = -125)

The exercise involves identifying that -125 is a perfect cube, as it is the cube of -5.

Evaluating the cube root of a number involves finding a value that, when cubed, returns the original number. The cube root notation is expressed as \( \sqrt[3]{x} \), which asks you to determine what number raised to the power of three gives you \( x \).
For example, to find \( \sqrt[3]{8} \), we need to determine a number \( n \) such that \( n^3 = 8 \). In this case, \( n = 2 \), because \( 2^3 = 8 \). Likewise, \( \sqrt[3]{-8} \) equals -2, since \( (-2)^3 = -8 \).
Cube root evaluation can involve both positive and negative numbers. Identifying perfect cubes simplifies the process and makes finding the cube root straightforward without delving into complex numbers. In the given exercise, since -125 is a perfect cube (\( (-5)^3 \)), the cube root \( \sqrt[3]{-125} \) simplifies neatly to -5.

Real numbers are numbers that can be found on the number line, including both positive and negative integers, fractions, and irrational numbers. When discussing cube roots, it’s important to distinguish which roots are real.
Cube roots of real numbers have a particular property: they always yield a real number, even when the radicand (the number inside the root) is negative. Unlike square roots, which can yield imaginary numbers if the radicand is negative, cube roots accept negative values and return a real result. This makes cube roots much more versatile in dealing with a broader range of real numbers.
For instance:

• The cube root of 8 (\( \sqrt[3]{8} \)) is 2, which is a real number.
• The cube root of -8 (\( \sqrt[3]{-8} \)) is -2, which is also a real number and perfectly valid.

In the exercise, \( \sqrt[3]{-125} \) is evaluated to -5, which confirms that the cube root of a negative number yields a real number, reinforcing that unlike square roots, cube roots of negative numbers are real and solvable within the realm of real numbers.