Problem 8 - Elementary and Intermediate Algebra

Question

Select the appropriate word to complete each of the following. If \(\sqrt[3]{x}\) is negative, then \(x\) must be _____.

Step-by-Step Solution

Answer

Negative

Step 1: Understanding the Cube Root

The cube root of a number, \(\frac{1}{3}\), is the value that, when raised to the power of 3, gives the original number. In this case, \(\frac{1}{3}\) of \(x\).

Step 2: Analyzing Negative Cube Roots

A cube root \(\frac{1}{3}\) can be negative because the cube of a negative number is still negative. For example, \(\frac{1}{3}(-8)=-2\) because \(-2^3=-8\).

Step 3: Conclusion

For the cube root \(\frac{1}{3}\) of \(x\) to be negative, \(x\) itself must be a negative number. This way, when raised to the power of 3, it retains its negative value.

Key Concepts

Negative Numbers

Negative numbers are those less than zero. They are represented with a minus sign (-) in front of the number. These numbers can affect various mathematical operations differently compared to positive numbers.

• Adding a negative number is like subtracting a positive. For example, 5 + (-3) equals 2.
• Subtracting a negative number is like adding a positive. So, 5 - (-3) equals 8.
• Negative numbers multiplied or divided by a positive number yield a negative result. For example, -4 * 2 equals -8.
• However, multiplying or dividing two negative numbers results in a positive number. So, -4 * -2 equals 8.

With cube roots, if the result is negative, the original number must be negative. This is because raising a negative number to an odd power (like 3) keeps the result negative. For instance, \(-2^3 = -8\). This means \(\frac{1}{3}{-8}=-2\).

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent denotes how many times the base is multiplied by itself. For instance, \(2^3 = 2 * 2 * 2=8\).
This concept forms the foundation for understanding cube roots and other types of roots.

• If the exponent is positive, it means repeated multiplication. So \a^3\ means \a*a*a\.
• If the exponent is zero, the result is always 1 (except when the base is zero).
• If the exponent is negative, it means the reciprocal raised to the positive exponent. So, \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).

Cube root itself represents the opposite operation of exponentiation to the power of three. To find the cube root of a number, you are trying to find a number which, when raised to the power of 3, gives the original number. For example, to find the cube root of 27, we ask which number raised to the power of 3 is 27. In this case, it is 3, since \(3^3 = 27\).

Algebraic Expressions

Algebraic expressions involve numbers, variables (like x or y), and arithmetic operations. They can range from simple to complex depending on the number of terms and operations involved.
Some key features of algebraic expressions include:

• Constants: Fixed values such as 5, -2, or 3.14.
• Variables: Symbols representing numbers that can change, typically letters like x or y.
• Coefficients: Numbers multiplying a variable. In 3x, the number 3 is the coefficient.
• Operators: Arithmetic actions like addition, subtraction, multiplication, and division.

When dealing with cube roots in algebra, it is important to simplify expressions to understand their properties better. First, recognize constant terms or coefficients. Second, identify and isolate variables if possible. For an expression like \(\frac{1}{3}{x}\), we analyze x based on known properties. If \(\frac{1}{3}{x}\) is negative, we conclude x must be negative. This helps in further algebraic manipulations and problem-solving.