Problem 51 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(\left(\frac{1}{2}\right)^{-3} = 8\)

Step 1: Understanding Negative Exponents

When dealing with negative exponents, we need to understand that they represent the reciprocal of the base raised to the positive exponent. For the given expression \(left(\frac{1}{2}\right)^{-3}\), the base \(\frac{1}{2}\) should be flipped, and the exponent of 3 should be applied.

Step 2: Taking the Reciprocal

Taking the reciprocal of \(\frac{1}{2}\) gives us \(2\). So the expression becomes \(2^{3}\).

Step 3: Applying the Exponent

Raise \(2\) to the power of 3. This means multiplying \(2\) by itself three times: \(2 \times 2 \times 2\).

Step 4: Calculating the Power

Calculate \(2 \times 2 = 4\), then multiply by \(2\) again: \(4 \times 2 = 8\). Factor each multiplication carefully to ensure accuracy.

Key Concepts

Reciprocal

When we talk about a reciprocal in mathematics, we mean the 'flip' of a fraction. If you start with a fraction, you reverse its numerator and denominator to find the reciprocal.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This concept is essential when working with negative exponents because a negative exponent signals that you'll be working with the reciprocal.

• The reciprocal of \(\frac{1}{2}\) becomes \(2\), since you swap the 1 (numerator) with the 2 (denominator).

Reciprocals help in transformations that simplify expressions. When converting negative exponents, understanding reciprocals becomes crucial.

Exponentiation

Exponentiation is the mathematical operation involving numbers called bases and exponents. In its simplest form, it's a way to represent repeated multiplication.
When you see an expression like \(x^n\), it means you'll multiply the base \(x\) by itself \(n\) times.
In our example, following the transformation into reciprocal form, we're left with \(2^3\). Here’s how it works step-by-step:

• Raise the base (\(2\)) to the power of \(3\): \(2 \times 2 \times 2\).
• This process helps in breaking down what seems like complex expressions into understandable steps.

It is vital because it is the foundational operation that transforms our simplified reciprocal into a final numerical value.

Simplification

Simplification in mathematics is all about reducing an expression to its minimal, simplest form. It's like cleaning up your math expression into a neat and tidy result that provides the answer clearly.
In our given exercise with the expression \(\left(\frac{1}{2}\right)^{-3}\), simplification involves several steps:

• First, understand the negative exponent to take the reciprocal, giving us \(2\).
• Then, apply the exponent by exponentiating \(2\): \(2^3\).
• Finally, calculate this to arrive at the simplified value of \(8\).

All these steps show the importance and beauty of simplification. When we simplify, we ensure that even complex-looking expressions become accurate and manageable solutions.