Problem 112 - Elementary Algebra

Question

For the following problems, evaluate each numerical expression. $$ 2^{-1} \cdot 3^{-1} \cdot 4^{-1} $$

Step-by-Step Solution

Answer

Question: Evaluate the expression $2^{-1} \cdot 3^{-1} \cdot 4^{-1}$. Answer: $\frac{1}{24}$

Step 1: Apply the rule for negative exponents

Rewrite the given expression using the rule for negative exponents: $$ 2^{-1} \cdot 3^{-1} \cdot 4^{-1} = \frac{1}{2^1} \cdot \frac{1}{3^1} \cdot \frac{1}{4^1} $$

Step 2: Simplify each term

Simplify each term by performing the exponentiation: $$ \frac{1}{2^1} \cdot \frac{1}{3^1} \cdot \frac{1}{4^1} = \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} $$

Step 3: Multiply the fractions

Multiply the fractions as follows: $$ \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} = \frac{1 \cdot 1 \cdot 1}{2 \cdot 3 \cdot 4} $$

Step 4: Simplify the final result

Simplify the final result by multiplying the numbers in the numerator and denominator: $$ \frac{1 \cdot 1 \cdot 1}{2 \cdot 3 \cdot 4} = \frac{1}{24} $$ So, the evaluated expression is: $$ 2^{-1} \cdot 3^{-1} \cdot 4^{-1} = \frac{1}{24} $$

Key Concepts

Exponentiation

Exponentiation is one of the fundamental operations in mathematics, often involving a base and an exponent. In the context of this exercise, the concepts of negative exponents play a key role. A negative exponent indicates that the base is on the wrong side of a fraction line. Rather than magnifying the base as positive exponents do, negative exponents invert the base. This can be understood with the rule:\( a^{-n} = \frac{1}{a^n} \).
For example, in the expression \( 2^{-1} \), the negative exponent implies a reciprocal, which becomes \( \frac{1}{2} \). When dealing with multiple negative exponents, like in the expression \( 2^{-1} \cdot 3^{-1} \cdot 4^{-1} \), each can be rewritten as \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \), respectively.

• Applying the negative exponent rule helps to convert seemingly complex expressions into fractions that are more manageable.
• Each term with a negative exponent results in a unit fraction.

Understanding these transformations is crucial for working accurately with mathematical expressions involving exponents.

Fraction Multiplication

Multiplying fractions is a straightforward process that involves two main steps: multiplying the numerators together and then multiplying the denominators together. In the problem at hand, once each term is set up as a fraction, you are ready to multiply:
\( \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} \).
Let's break down the multiplication:

• The numerators are: \( 1 \times 1 \times 1 \ = 1 \).
• The denominators are: \( 2 \times 3 \times 4 \ = 24 \).

This results in the fraction \( \frac{1}{24} \).
Key tips for fraction multiplication:

• The process doesn't change based on the size of numbers; it's always about following the same straightforward steps.
• Keep fractions in the simplest form unless directed otherwise.

Mastering fraction multiplication helps handle more intricate mathematical operations with ease.

Simplifying Expressions

Simplifying expressions involves several techniques, including applying rules for exponents, reducing fractions, and sometimes factoring. In this exercise, once the multiplication of fractions is done, the resulting expression \( \frac{1}{24} \) is already in its simplest form because \( 1 \) and \( 24 \) have no common factors besides \( 1 \).

Here are some strategies used in simplification:

• Ensure all calculations, like exponentiation and fraction multiplication, are performed fully before deciding if simplification is complete.
• Be aware of potential common factors that could further reduce a fraction.
• Review each stage of your work to confirm no further simplification is possible.

Simplifying helps create a clearer, more understandable result, critical for ensuring accuracy in calculations and presentations.