Problem 30 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(\frac{2^{-4}}{3^{-2}}\right)^{-2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{256}{81}\).

Step 1: Apply Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Using this rule, rewrite \(2^{-4}\) as \(\frac{1}{2^4}\) and \(3^{-2}\) as \(\frac{1}{3^2}\). Therefore, the expression becomes: \[\left(\frac{\frac{1}{2^4}}{\frac{1}{3^2}}\right)^{-2}\]

Step 2: Simplify the Fraction

When dividing fractions, multiply by the reciprocal. So, \(\frac{\frac{1}{2^4}}{\frac{1}{3^2}}\) becomes \(\frac{3^2}{2^4}\). The whole expression is now:\[\left(\frac{3^2}{2^4}\right)^{-2}\]

Step 3: Reapply the Negative Exponent

The expression \(\left(\frac{3^2}{2^4}\right)^{-2}\) can be simplified using the negative exponents rule, which results in:\[\left(\frac{2^4}{3^2}\right)^2\]

Step 4: Distribute the Exponent

Raise both the numerator and the denominator of the fraction \(\left(\frac{2^4}{3^2}\right)^2\) to the power of 2:\[\frac{(2^4)^2}{(3^2)^2}\]This results in:\[\frac{2^8}{3^4}\]

Step 5: Simplify the Powers

Calculate the powers in the fraction:- \(2^8 = 256\)- \(3^4 = 81\)Now substitute back these values:\[\frac{256}{81}\]

Step 6: Final Simplification

Since \(\frac{256}{81}\) is already simplified as the greatest common divisor of 256 and 81 is 1, the expression remains:\[\frac{256}{81}\]

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means ensuring that the numerator and the denominator have no common factors other than 1. In mathematical expressions, simplifying fractions helps in easier calculations and a better understanding of the expression.
To simplify a fraction, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• Ensure the fraction is in its simplest form where further division is not possible.

In the given expression, we deal with fractions during the simplification process, such as turning complex expressions into a form where numerators like 2^8 and denominators like 3^4 are involved. Once calculated, 256 and 81 have no common factors, confirming that the expression \(\frac{256}{81}\) is already simplified.

Reciprocal

The reciprocal is a critical mathematical concept, especially when dealing with fraction operations. A reciprocal of a number or a fraction is what you multiply it by to get the multiplication identity, which is 1.
For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When performing division involving fractions, reciprocals play a vital role. Multiplying by a reciprocal transforms division into multiplication.
In our expression, the transition from \(\frac{\frac{1}{2^4}}{\frac{1}{3^2}}\) to \(\frac{3^2}{2^4}\) uses the concept of reciprocals. By changing the division of fractions into multiplication by a reciprocal, we simplify complex fraction division.

Exponentiation

Exponentiation involves raising numbers to a power, which signifies multiplying the base by itself a specific number of times. The base is the number being multiplied, and the exponent is the number of times it's multiplied.
Negative exponents, such as in 2\(^{-4}\), imply reciprocals: \(a^{-n} = \frac{1}{a^n}\). Thus, using exponentiation here involves converting negative exponents to positive ones by using their reciprocal forms.
Further in the solution, the application of exponentiation arises again when dealing with entire fractions raised to a power. After simplifying the expression to \(\frac{2^4}{3^2}\), both the numerator and denominator are raised to the power of 2, giving us \(\frac{(2^4)^2}{(3^2)^2}\). This results in 2 raised to the power of 8, and 3 to the power of 4.

Numerical Expression Simplification

Numerical expression simplification is about making an expression as easy to understand and compute as possible. It involves different mathematical rules and operations such as fraction simplification, reciprocal calculation, and exponentiation.
The goal of simplification is to reach a single, simple numeric result or an irreducible fraction, reducing computational complexity. In this exercise, simplification step-by-step ensures that long and complex numerical expressions are expressed in terms of easier numbers like 256 and 81.
In the final stages of the exercise, after calculating the powers 2^8 and 3^4, substitute these values back into the fraction. We find that since the division cannot be simplified further with common divisors, the expression is as simple as it can get. Thus, the expression \(\frac{256}{81}\) is the simplest possible form of the original problem.