Problem 9 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$(-4)^{-3}$$

Step-by-Step Solution

Answer

The simplified expression of \\((-4)^{-3}\\) is \\frac{1}{-64}\\.

Step 1: Understanding Negative Exponents

When you encounter a negative exponent like \((-4)^{-3}\), it suggests that you have the reciprocal of the base raised to the positive of that exponent. For instance, \((x^{-a}) = \frac{1}{x^{a} }\)."

Step 2: Applying Negative Exponent Rule

Following the negative exponent rule, \((-4)^{-3}\) changes to \[\left( -4 \right)^{-3} = \frac{1}{(-4)^{3}}\] . This means you can write the expression with a positive exponent in the denominator.

Step 3: Calculate the Positive Exponent

Now, calculate \((-4)^{3}\). This means multiplying \((-4) \times (-4) \times (-4)\). First, \((-4) \times (-4) = 16\). Then, \(16 \times (-4) = -64\). This results in \((-4)^{3} = -64\)."

Step 4: Write the Fraction

Substitute \((-4)^{3} = -64\) back into the fraction: \[\frac{1}{-64}\]. This is the simplification of the original expression.

Key Concepts

Understanding Reciprocals

The term 'reciprocal' might sound complex, but it just means flipping a number. If you have a number like 5, its reciprocal is \( \frac{1}{5} \). Basically, you're making it a fraction.
If the number is a fraction, like \( \frac{3}{4} \), the reciprocal would be \( \frac{4}{3} \). When you multiply a number by its reciprocal, you always get 1. That's the magic of reciprocals!
Reciprocals are particularly helpful when dealing with negative exponents. For instance, with an expression like \(x^{-a}\), taking the reciprocal lets you flip it to a positive exponent: \( \frac{1}{x^a} \). This trick makes complex calculations simpler.

Positive Exponents Explained

Positive exponents are like repeat instructions. They tell you how many times to multiply a number by itself. For example, if you see \(3^2\), it means \(3 \times 3 = 9\).
Let's dive into some key points about positive exponents:

• Always lead to multiplication: \( (2^3) = 2 \times 2 \times 2 \).
• Can apply to any number, whole or decimal, like \( (0.5)^2 \).

In contrast to negative exponents, positive exponents don't involve reciprocals. Once you convert a negative exponent using a reciprocal, the computations involve just multiplying as indicated by the positive exponent.

Simplification Process

Simplification in math means making an expression as simple as possible. For negative exponents, simplification usually involves converting them using reciprocals first.
Consider the example \((-4)^{-3}\). First, rewrite it using the reciprocal concept: \( \frac{1}{(-4)^3} \).
Next, calculate the positive exponent part \((-4)^3\):

• Multiply \((-4) \times (-4) = 16\).
• Then, multiply \(16 \times (-4) = -64\).

This gives us \((-4)^3 = -64\), leading to the simplified expression \(\frac{1}{-64}\). The result is a simpler fraction that represents the original expression in its most straightforward form.