Problem 41 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(-\frac{4}{x}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-\frac{64}{x^{3}}.\)

Step 1: Identify the Base and Exponent

In the expression \(-\frac{4}{x}^{3}\), the base is \(-\frac{4}{x}\) and the exponent is 3.

Step 2: Apply the Power to the Numerator and Denominator

Put the base of -4 and the denominator x to the power of 3. Remember, when a fraction is raised to a power, both the numerator and the denominator are raised to that power. This gives us \[-\left(4^{3}\right)/\left(x^{3}\right).\]

Step 3: Perform the Exponentiation

Calculate the exponentiation to find that \(4^{3} = 64\) and \(x^{3}\) remains as is, because we don't know the value of x. So, the expression simplifies to \[-\frac{64}{x^{3}}.\]

Key Concepts

Exponential Expression

When we talk about an exponential expression, we're referring to a mathematical notation that involves a base and an exponent. The expression normally appears in the form of a base raised to the power of an exponent (e.g., a^n). In plain language, this tells us how many times to multiply the base by itself. For example, 3^4 (read as 'three to the fourth power' or 'three raised to the power of four') means you multiply 3 by itself 4 times: 3 \times 3 \times 3 \times 3, which equals 81.

In our exercise, the expression given is \(\left(-\frac{4}{x}\right)^{3}\). The objective is to simplify this expression to a form that's easier to understand or work with in later operations. Simplifying an exponential expression can include performing exponentiation, as well as applying any necessary operations such as multiplication or division to the base and exponent.

Base and Exponent

Understanding the roles of the base and exponent is crucial for working with exponential expressions. The base is the value that is being multiplied repeatedly, and the exponent tells us the number of times the multiplication occurs. In the provided exercise, \(-\frac{4}{x}\) is the base, which means it's the value to be multiplied by itself, and 3 is the exponent indicating that the multiplication will happen three times.

In the case of our exercise, to simplify \(-\frac{4}{x}\)^3, we apply the exponent to both the numerator and denominator of the fraction separately. The key is to raise each part of the base to the power of the exponent independently. Thus, we end up with \(-4^3\) and \(x^3\) and then perform the exponentiation for the numerical part, eventually arriving to the simplified form \(-\frac{64}{x^3}\).

Negative Exponents

The concept of negative exponents introduces an additional twist in working with exponential expressions. A negative exponent tells us to take the reciprocal of the base and then apply the positive exponent. For instance, \(x^{-n}\) is equal to \(\frac{1}{x^n}\). It's important to remember that a negative exponent does not make the whole number negative; instead, it changes the position of the base from the numerator to the denominator or vice versa.

In the context of our textbook exercise, we aren't directly dealing with negative exponents, but it's good to be aware of them. When simplifying expressions with negative exponents, one would start by converting them to positive exponents and then simplify as usual. For example, \((-3)^{-2}\) can be rewritten as \(\frac{1}{(-3)^2} = \frac{1}{9}\), demonstrating how we move the base to the denominator and remove the negative sign from the exponent.