Problem 27 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \frac{1}{4^{-4}} $$

Step-by-Step Solution

Answer

The evaluation of the expression \( \frac{1}{4^{-4}} \) is 256.

Step 1: Understand the formula

Firstly, recognize the form of the expression. It is \( \frac{1}{4^{-4}} \). This is a fraction where the denominator is 4 to the power -4 derivative.

Step 2: Evaluate the denominator

The denominator is \(4^{-4}\). A negative power means the reciprocal of the base to the positive power. So, \(4^{-4}\) is equivalent to \(\frac{1}{4^{4}}\). So, it simplify to \(\frac{1}{256}\).

Step 3: Calculate the final value

So the expression will become \( \frac{1}{ \frac{1}{256}} \). While dividing fractions, we multiply the numerator by the reciprocal of the denominator, so, the expression simplify to \(1*256\), which gives us 256 as final result.

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they are quite simple once you get the hang of them. When you see a negative exponent, it is telling you to take the reciprocal (or flip) of the number and then apply the positive exponent. This means if you have something like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).
Here's how it works:

• Turn the number into its reciprocal.
• Change the exponent from negative to positive.

In the exercise we are working on, we have \( 4^{-4} \). By applying the rule on negative exponents:

• Take the reciprocal of 4, which becomes \( \frac{1}{4} \).
• Then, change the \(-4\) to a positive 4, giving \( \frac{1}{4^4} \).

This simplifies \( 4^{-4} \) into \( \frac{1}{4^4} \), making it much easier to evaluate.

Fractional Exponents

Fractional exponents might also seem a little confusing at first, but they offer a powerful way to express roots using exponents. They are another versatile part of algebraic expressions, representing roots. A fractional exponent like \( a^{\frac{m}{n}} \) can be understood as a root: \( \sqrt[n]{a^m} \).
Let's break it down:

• The numerator of the fraction (m) is the power you'd raise the number to, once you've taken its nth root.
• The denominator of the fraction (n) represents the root.

For example, \( 8^{\frac{1}{3}} \) can be interpreted as the cube root of 8. Since the cube root of 8 is 2, \( 8^{\frac{1}{3}} = 2 \). It’s all about thinking about both the power and the root together. While fractions and roots aren't directly used in the main exercise provided, knowing how fractional exponents work helps reinforce understanding of exponent rules.

Reciprocals

A reciprocal is simply flipping a number over. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). When we talk about reciprocals, all we're really referring to is this switching of positions between the numerator and denominator.
Reciprocals are especially useful when dividing fractions. According to the division rule of fractions:

• You multiply by the reciprocal of the divisor.

In the context of the original problem, when you encounter \( \frac{1}{\frac{1}{256}} \), it simplifies to 256 by finding the reciprocal and then multiplying. This approach underlies how negative exponents work as well, since \( a^{-1} \) is essentially conceptually similar to finding the reciprocal of \( a \). It’s another example of how different algebraic concepts are tied together, and understanding this link can enhance your mathematical reasoning.