Problem 9 - Intermediate Algebra

Question

Simplify each numerical expression. \(\left(-\frac{1}{2}\right)^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is \(-8\).

Step 1: Understand Negative Exponent Rule

When simplifying expressions with negative exponents, the rule is that you take the reciprocal of the base and then change the exponent to positive. In this case, the expression is \( \left(-\frac{1}{2}\right)^{-3} \). According to the rule, this becomes \( \left(-\frac{1}{2}\right)^{-3} = \left(-2\right)^{3}\).

Step 2: Apply Positive Exponent

Now that we have \( (-2)^{3} \), we need to calculate this expression. This means multiplying -2 by itself three times: \( (-2) \times (-2) \times (-2) \).

Step 3: Perform Multiplication

Calculate \( (-2) \times (-2) \times (-2) \): 1. First, multiply \( (-2) \times (-2) = 4 \).2. Then multiply the result by \(-2\) again: \( 4 \times (-2) = -8 \).

Key Concepts

Negative Exponent Rule

Understanding negative exponents is a crucial part of simplifying expressions. When you see a negative exponent, it's a signal to find the reciprocal of the base. This means turning the base upside-down, or swapping the numerator and the denominator if it's a fraction.
For instance, the expression \( \left(-\frac{1}{2}\right)^{-3} \) uses the negative exponent rule. To simplify, take the reciprocal of \( -\frac{1}{2} \) to get \( -2 \). After you take the reciprocal, you flip the sign of the exponent to get a positive exponent, resulting in \( \left(-2\right)^{3} \).
In general, the negative exponent rule can be expressed as:

• \( a^{-n} = \frac{1}{a^{n}} \) - For any non-zero \( a \) and positive integer \( n \).
• The process turns a division into a multiplication by inverting the base.

Remember, a negative exponent does not make the base negative; it simply inverts it.

Reciprocal

A reciprocal refers to the flipped version of a fraction. You swap the numerator and the denominator. It's like saying: "If you're up, go down; if you're down, go up." This concept is especially handy when dealing with negative exponents in math.
For example, the reciprocal of \( -\frac{1}{2} \) is \( -2 \). Essentially, you switch places of 1 and 2 while keeping the negative sign upfront.
Key points to remember about reciprocals:

• The reciprocal of a number \( x \) is \( \frac{1}{x} \).
• Multiplying a number by its reciprocal always gives 1: \( x \times \frac{1}{x} = 1 \).
• If your number is already negative, the sign doesn't change when taking the reciprocal.

Think of the reciprocal as a mirror reflection in math, helping you to simplify expressions effectively.

Multiplication of Negative Numbers

Multiplying negative numbers follows specific rules just like any other numbers. When you multiply negatives, the signs play a pivotal role in determining the final result.
Consider the expression \( (-2) \times (-2) \times (-2) \), part of simplifying \( \left(-\frac{1}{2}\right)^{-3} \).
Here's a simple breakdown:

• When two negative numbers are multiplied, the result is positive: \( (-2) \times (-2) = 4 \).
• If you then multiply this positive result (4) by another negative number (-2), the result becomes negative again: \( 4 \times (-2) = -8 \).
• So, an odd number of negative signs results in a negative product.

Always apply these rules:

• Negative \( \times \) Negative = Positive
• Negative \( \times \) Positive or Positive \( \times \) Negative = Negative

Understanding these rules ensures clarity in tackling multiplication with negative numbers.