Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left(-a^{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{a^6} \).

Step 1: Identify the given expression

We are given the expression: \[ \left(-a^{2}\right)^{-3} \]

Step 2: Apply the exponent-properties

Using the property \( (ab)^n = a^n b^n \), we can simplify the expression as follows: \[ \left(-a^{2}\right)^{-3} = (-1)^{-3} \cdot a^{-6} \]

Step 3: Convert negative exponents to positive exponents

Using the property \( a^{-n} = \frac{1}{a^n} \), we can convert the negative exponents to positive exponents: \[ (-1)^{-3} \cdot a^{-6} = \frac{1}{(-1)^3} \cdot \frac{1}{a^6} \]

Step 4: Simplify the expression

Now, we can simplify the expression further by evaluating the exponent in the numerator: \[ \frac{1}{(-1)^3} \cdot \frac{1}{a^6} = \frac{1}{-1} \cdot \frac{1}{a^6} \]

Step 5: Write the final expression

Finally, we can write the simplified expression with positive exponents only: \[ \frac{-1}{a^6} \] The simplified expression is \( \frac{-1}{a^6} \).

Key Concepts

Understanding Negative Exponents

The concept of negative exponents can be initially confusing, but they are an essential part of algebra. When you see an expression like \( a^{-n} \), it indicates the reciprocal of the base raised to a positive exponent. In simpler terms, \( a^{-n} = \frac{1}{a^n} \). This transformation is an application of one of the most basic exponent properties.

When you come across negative exponents in an expression, you rewrite them as the reciprocals with positive exponents to simplify the expression. For instance, \( (-a^{2})^{-3} \) becomes \( (-1)^{-3} \cdot a^{-6} \), and each part of this product is turned into the reciprocal with a positive exponent. Negative exponents signify division, just as positive exponents denote multiplication.

Exponent Properties

Exponents are not just about making numbers larger. They follow specific rules or properties that allow us to manipulate and simplify expressions in a systematic way. One key property is that of an exponent applied to a product: \( (ab)^n = a^n b^n \). This property allows us to separate the individual components of an expression raised to a power.

Another central property deals with exponents of exponents, where \( (a^n)^m = a^{n \cdot m} \). Applying these properties makes it easier to simplify expressions before evaluating them. It's essential to remember these properties as they provide the tools to unravel more complicated exponential expressions into simpler, more manageable parts.

Mathematical Simplification

The ultimate goal of mathematical simplification is to rewrite expressions in the most accessible format possible. Simplification may involve factoring, combining like terms, or applying mathematical properties like those of exponents. In the exercise \( (-a^{2})^{-3} \), simplification consists of applying exponent rules to first separate and then convert negative exponents to positive exponents, leading us to \( \frac{1}{(-1)^3} \cdot \frac{1}{a^6} \).

Finally, evaluating any constants, as \( (-1)^3 = -1 \), simplifies the expression to \( \frac{-1}{a^6} \). The choice of steps in the simplification process influences how quickly and efficiently we can reach the simplified form. It's important to apply exponent properties systematically to avoid common errors and achieve the correct simplified form.