Problem 2 - Algebra 2 and Trigonometry

Question

If \(a\) is a negative number, is \(-\sqrt[3]{-8 a^{3}}\) a positive number, a negative number, or not a real number? Explain your answer. oning Skills.

Step-by-Step Solution

Answer

\(-\sqrt[3]{-8 a^{3}}\) is a negative number since \(a\) is negative.

Step 1: Understanding the cube root

The cube root of a number is that number which, when cubed, gives the original number. The cube root is also valid for negative numbers. Thus, \(\sqrt[3]{-8 a^{3}}\) can be further simplified.

Step 2: Simplifying the cube root expression

Simplify the expression \(\sqrt[3]{-8 a^{3}}\). First, factor \(-8 a^3\) as \((-1) imes (8) imes (a^3)\). The cube root of each component gives: \(\sqrt[3]{-1} = -1\), \(\sqrt[3]{8} = 2\), and \(\sqrt[3]{a^3} = a\). So, \(\sqrt[3]{-8 a^3} = -1\times 2\times a = -2a\).

Step 3: Analyzing the prefactor

The expression is \(-\sqrt[3]{-8 a^{3}} = -(-2a) = 2a\). Basically, when you apply the cube root and the negative sign outside, you end up with \(2a\).

Step 4: Determining the sign of the final expression

Characterize the sign of \(2a\): since \(a\) is given as a negative number, \(2a\) (¨two times a negative number) will also be negative.

Key Concepts

cube root of negative numbers

The cube root of a number, denoted as \( \sqrt[3]{x} \), is a value that, when multiplied by itself three times, gives the original number \( x \). This concept is crucial in mathematics, especially when dealing with negative numbers. Unlike square roots, cube roots of negative numbers are viable and result in a real number.

For example, the cube root of \( -8 \) is \( -2 \) because \( (-2) \times (-2) \times (-2) = -8 \). Cubing any negative number results in a negative product. Therefore, the cube root of a negative number yields a negative result. This makes the cube root operation particularly unique because it maintains the sign of the original number, allowing calculations to stay in the realm of real numbers.

simplifying expressions

Simplifying expressions involves breaking down a complex mathematical expression into simpler, more manageable parts. This helps in understanding and solving problems more effectively.

Let's take the expression \( \sqrt[3]{-8 a^3} \). It seems complex at first, but breaking it down simplifies things. Begin by factoring it into manageable parts:

• Factor \(-8 a^3\) as \( (-1) \times 8 \times a^3 \).
• Find the cube root of each part: \( \sqrt[3]{-1} = -1 \), \( \sqrt[3]{8} = 2 \), and \( \sqrt[3]{a^3} = a \).

By multiplying the simplified components: \(-1 \times 2 \times a = -2a\). This simplification process makes it easier to understand the behavior of the expression and proceed with solving the problem.

negative numbers

Negative numbers are integral in mathematics, representing values less than zero. When dealing with negative numbers in expressions, understanding their properties helps in predicting the behavior of the resulting expressions.

In our context, consider \( 2a \) where \( a \) is a negative number. When you multiply a positive number \( 2 \) with a negative number, the result is negative. Thus, even though the expression \( -\sqrt[3]{-8 a^{3}} \) simplifies to \( 2a \), the product remains negative due to the initial condition that \( a \) is negative.

Being aware of these traits of negative numbers simplifies the understanding and manipulation of mathematical expressions, especially when predicting the final sign of an expression after various operations are applied.