Problem 106 - Algebra A Combined Function

Question

$$ \text { Explain why } \sqrt[3]{-64} \text { is a real number. } $$

Step-by-Step Solution

Answer

\( \sqrt[3]{-64} = -4 \), which is a real number because a cube root of a negative number is still real.

Step 1: Understanding Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In the context of real numbers, cube roots can be applied to both positive and negative numbers.

Step 2: Calculate the Cube Root of -64

To find \ \( \sqrt[3]{-64} \), we need to identify a number that, when multiplied by itself three times, results in -64. Let's consider -4. Compute \ \((-4) \times (-4) \times (-4) \):\ -4 \times -4 = 16\ 16 \times -4 = -64 \ Therefore, \( \sqrt[3]{-64} = -4 \).

Step 3: Verify that Cube Roots Can Be Negative

It's important to remember that unlike square roots, which must result in non-negative values when considering just real numbers, cube roots can be negative. This stems from the fact that a negative number multiplied by itself an odd number of times (such as three) will yield a negative result.

Step 4: Conclusion

Since \( -4 \times -4 \times -4 = -64 \), and cube roots can be negative, \( \sqrt[3]{-64} = -4 \) is indeed a real number.

Key Concepts

Understanding Real Numbers

Real numbers are the building blocks of almost all the numbers we encounter in our daily lives. They include:

• Whole numbers (such as 0, 1, 2,...)
• Fractions
• Decimals
• Negative numbers

Essentially, every number that can be found on the number line is a real number. In mathematics, real numbers encompass both rational numbers (like fractions) and irrational numbers (which cannot be expressed as simple fractions, such as \(\sqrt{2}\) or \(\pi\)).

Negative numbers are also real numbers. They are any numbers less than zero and are often found in contexts involving debts or temperatures below freezing. When considering operations like cube roots, both positive and negative numbers are relevant because the cube root of a negative number can also be a negative number itself. This characteristic is unique to real numbers and does not apply to other number systems like complex numbers.

Exploring Negative Cube Roots

Understanding cube roots, especially of negative numbers, is simpler once you realize how multiplication works with negatives. A cube root of a number is a value that, when cubed (or multiplied by itself twice), results in the original number. For positive numbers, this is straightforward: what is the cube root of 8? It's 2, since \(2 \times 2 \times 2 = 8\).

When dealing with negative numbers, such as \(-64\), the approach is similar. We are looking for a number that gives us \(-64\) when raised to the power of three. Consider \(-4\):

• \(-4 \times -4\) results in \(16\), because the multiplication of two negative numbers is positive.
• Multiplying \(16\times -4\) gives \(-64\).

Thus, the cube root of \(-64\) is \(-4\). This outcome is possible because a negative number multiplied an odd number of times (such as three times) will remain negative.

Hence, unlike square roots, which in the realm of real numbers always yield non-negative results, cube roots can indeed yield negative results, aligning with the properties of real numbers.

The Role of Multiplication in Cube Roots

Multiplication plays a crucial role when finding cube roots, especially because cube roots are essentially the reverse process of cubing. To multiply means to add a number repeatedly a certain number of times. With negative numbers, the rules slightly different from positive numbers:

• Multiplying two positive numbers yields a positive number.
• Multiplying two negative numbers also results in a positive number.
• Multiplying a positive and a negative number results in a negative number.

When we find the cube root of a negative number, we rely on these rules to determine the correct solution. For instance, with \(\sqrt[3]{-64}\), multiplication helps verify our solution of \(-4\):

• First multiplication gives us a positive: \(-4 \times -4 = 16\).
• Next, it changes sign because \(16\times -4 = -64\).

This verification through multiplication ensures that the number we identified is indeed the cube root because it successfully returns the original number when multiplied by itself three times.