Problem 8 - 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 the Negative Exponent Rule

The negative exponent rule states that when you have a negative exponent, you need to take the reciprocal (or invert the fraction) and change the exponent to a positive. So, for \(\left(\frac{1}{2}\right)^{-3}\), the base fraction should be inverted.

Step 2: Invert the Fraction

To invert \(\frac{1}{2}\), you turn it upside down, so it becomes \(2\). Therefore, \(\left(\frac{1}{2}\right)^{-3}\) becomes \(2^3\).

Step 3: Apply the Positive Exponent

Now that the base is \(2\), raise it to the power of \(3\): \[2^3 = 2 \times 2 \times 2 = 8\]Thus, \(2^3 = 8\).

Key Concepts

Numerical Expressions

In mathematics, numerical expressions are combinations of numbers and mathematical operations such as addition, subtraction, multiplication, and division. They are essential because they help us represent and solve problems in numerical form. The goal is to simplify these expressions to find a single value.

When dealing with numerical expressions involving exponents, the process is similar but includes the additional operation of raising a number to a power. For example, in an expression like \(\left(\frac{1}{2}\right)^{-3}\), it's crucial to follow the rules of exponents to simplify.

• The expression uses a fraction \(\frac{1}{2}\) raised to a negative power, \(-3\).
• The negative exponent indicates a reciprocal inversion of the base fraction.
• After inversion, apply the positive exponent to find the solution.

Understanding this process is essential for handling more complex mathematical problems involving exponents and other operations.

Reciprocal

A reciprocal is essentially the upside-down version of a fraction. For any non-zero number, the reciprocal is the number that, when multiplied by the original number, results in 1. In fraction form, the reciprocal is achieved by swapping the numerator and the denominator.

Consider the fraction \(\frac{1}{2}\):

• Its reciprocal is 2 because \(\frac{1}{2} \times 2 = 1\).

Reciprocals play a crucial role when dealing with negative exponents. According to the negative exponent rule, a base with a negative exponent requires converting that base to its reciprocal and changing the exponent to positive. For \(\left(\frac{1}{2}\right)^{-3}\), the process involves converting \(\frac{1}{2}\) to its reciprocal, 2, resulting in \(2^3\).

This simple step ensures that calculations involving negative exponents can be performed as straightforward operations involving positive powers.

Exponent Rules

Exponent rules are fundamental guidelines that dictate how to handle powers in mathematical expressions. These rules simplify complex calculations and ensure consistency when working with exponents of any kind:

• The Product Rule involves multiplying powers with the same base by adding their exponents: \(a^m \times a^n = a^{m+n}\).
• The Quotient Rule deals with dividing powers with the same base: \(\frac{a^m}{a^n} = a^{m-n}\).
• The Power Rule involves raising a power to another power: \((a^m)^n = a^{m \times n}\).
• The Negative Exponent Rule changes a negative exponent to a positive one by inverting the base: \(a^{-n} = \frac{1}{a^n}\).

When faced with a negative exponent as in \(\left(\frac{1}{2}\right)^{-3}\), the negative exponent rule necessitates taking the reciprocal of \(\frac{1}{2}\), transforming it into 2, and then raising 2 to the positive exponent, 3. Sequential application of these rules helps simplify numerical expressions accurately and efficiently.