Problem 9 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(-8\).

Step 1: Understand Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, \((-\frac{1}{2})^{-3}\) means we take the reciprocal of \(-\frac{1}{2}\) and raise it to the power of 3.

Step 2: Find the Reciprocal of the Base

The reciprocal of \(-\frac{1}{2}\) is \(-2\). This is because to find the reciprocal, you switch the numerator and the denominator of the fraction.

Step 3: Raise the Reciprocal to the Positive Exponent

Now, compute \((-2)^3\). This means multiplying \(-2\) by itself three times: \(-2 \times -2 \times -2\).

Step 4: Perform the Multiplication

Calculate \(-2 \times -2 = 4\) and then multiply by \(-2\) again: \(4 \times -2 = -8\). Thus, \((-2)^3 = -8\).

Step 5: Simplified Expression

The simplified form of \((-\frac{1}{2})^{-3}\) is \(-8\), as calculated above.

Key Concepts

Understanding the Reciprocal

When simplifying expressions with negative exponents, getting familiar with the concept of a reciprocal is crucial. The reciprocal of a number or a fraction is simply flipping the numerator and the denominator. For instance, the reciprocal of \(-\frac{1}{2}\) is \(-2\). Think of it as turning the number upside down when it's in fraction form. This step is vital because negative exponents essentially mean you will be working with the reciprocal of the base.

Remember, every number and every fraction has a reciprocal. For a simple whole number like 5, its reciprocal is \(\frac{1}{5}\). In contrast, for a fraction like \(-\frac{1}{2}\), you can find its reciprocal by swapping the positions of the numerator and the denominator.

Simplifying a Numerical Expression

A numerical expression is a mathematical phrase involving numbers and operation symbols, but no variables. In our exercise, the expression \((-\frac{1}{2})^{-3}\) is an example of this. Simplifying such expressions often involves several steps: understanding what each part of the expression means, and then performing arithmetic operations according to the rules of mathematics.

You begin by identifying any negative exponents, which signal the need to use the reciprocal of the base. From here, you execute the necessary exponentiation or other operations in sequence. Breaking it down into smaller, clear steps makes the calculation much more manageable and less prone to errors.

The Power of Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of the exponent. This represents repeated multiplication of the base. For example, in \(2^3\), \2\ is the base and \3\ is the exponent, which means \2 \times 2 \times 2\.

When dealing with a negative base like in \(-2)^3\), it’s important to consider the effect of the exponent on the sign. Odd powers maintain the sign of the base, hence \(-2 \times -2 \times -2 = -8\). Recognizing this pattern is extremely useful when solving problems involving negative numbers and exponentiation.

• A positive exponent implies straightforward repeated multiplication.
• A negative exponent requires finding the reciprocal first, then applying the exponent as usual.

Understanding these concepts helps make exponentiation problems, even with negative numbers, much simpler to grasp.