Question

If \(x<0\), which of the following is true? C \(x^3\) is positive. ( \(x^3\) is negative. \(x^2\) could be either positive or negative. I don't know.

Short answer

The correct answer is \(x^3\) is negative.

Step-by-step solution

Analyzing the Choices

Looking at all the given choices. Evaluating each situation.

Checking the Cube of Negative Number

In algebra, when any negative number is raised to the power of an odd number. The result will also be negative. In this case, \(x^3\) will also be negative for \(x<0\).

Checking the Square of Negative Number

In algebra, when a negative number is squared, or raised to any other even power, the result will be positive. In this case, \(x^2\) will be positive for \(x<0\).

Key concepts

Negative Numbers

Negative numbers are those that are less than zero. They are represented with a minus sign in front of them like -2, -5, or -10.
In an algebraic context, dealing with negative numbers often involves understanding how they interact with other mathematical operations.

• Addition: When adding a negative number, it is like subtracting its positive counterpart. For example, 3 + (-2) = 1.
• Subtraction: Subtracting a negative number means adding its absolute value. For instance, 7 - (-3) = 7 + 3 = 10.
• Multiplication and Division: When multiplying or dividing two negative numbers, the result is positive. However, if only one number is negative, the result is negative.

Understanding negative numbers paves the way for solving problems involving exponents and powers effectively.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The exponent tells you how many times to multiply the number, or base, by itself.
For example, in the expression 3^4, 3 is the base and 4 is the exponent. This means 3 is multiplied by itself 4 times, equating to 3 * 3 * 3 * 3 = 81.
It is critical to understand how exponents work with negative numbers:

• When the base is negative, such as in (-3)^2, the result can vary depending on whether the exponent is even or odd.
• If the base is positive, the result remains positive regardless of the exponent.

Exponents simplify expressions and help in understating the behavior of numbers under repeated multiplication, essential for mastering algebra.

Even Powers

Even powers occur when a number is raised to an even exponent, such as 2, 4, or 6. When using even powers with negative numbers, interesting results happen.

• For example, (-3)^2 = (-3) * (-3) = 9. The negative sign is eliminated because the negative base multiplies an even number of times, resulting in a positive product.

This property holds not just for squaring but for any even power:

• If the base is negative, raising it to an even power results in a positive number.
• If the base is positive, even powers simply affirm the positivity.

This concept is crucial as it counters the simplification errors that might come when handling expressions with different signs and exponents.

Odd Powers

Odd powers happen when a number is raised to an odd exponent, like 1, 3, or 5. This situation is markedly different from even powers when applied to negative numbers.

• Take (-2)^3, for instance: (-2) * (-2) * (-2) = -8. Here, the result is negative because the negative number is multiplied an odd number of times.

Odd powers retain the original sign of the base:

• If the base is negative, the result after using an odd exponent is negative.
• If the base is positive, the result remains positive for any odd exponent.

Understanding odd powers is essential to master various algebraic problems and helps in predicting the outcome of expressions involving negative bases and odd exponents.