Problem 56 - Elementary and Intermediate Algebra

Question

Simplify. \(\left(\frac{2}{t}\right)^{-4}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{t^4}{16}\).

Step 1: Understand Negative Exponent

A negative exponent like \(-n\) indicates taking the reciprocal and then raising it to the positive \(n\). So, we first write the expression as the reciprocal, which is \((t/2)^4\).

Step 2: Apply the Exponent

Now, we apply the exponent to both the numerator and the denominator: \((t/2)^4 = (t^4)/(2^4)\).

Step 3: Simplify the Numerator and Denominator

Calculate \(t^4\) for the numerator, which stays \(t^4\) as it is. Calculate \(2^4\) for the denominator, which is \(16\).

Step 4: Create the Final Expression

Combine the simplified numerator and denominator to get the final expression, which is \((t^4)/(16)\).

Key Concepts

Understanding Negative Exponents

When you encounter a negative exponent, it's essential to remember that it means to take the reciprocal of the base number and then apply the positive exponent.
This is because a negative exponent indicates the opposite of multiplying repeatedly; it essentially turns multiplication into division. For example, when you see something like \(a^{-n}\), interpret it as \(\frac{1}{a^n}\). So in the exercise we began with, \(\left(\frac{2}{t}\right)^{-4}\), the negative exponent tells us to flip the fraction \(\frac{2}{t}\) into \(\frac{t}{2}\) before raising it to the 4th power.
This reciprocal process is a core method in algebra for handling negative exponents, and mastering this concept is crucial for simplifying such expressions effectively.

The Basics of Fractions

Fractions are mathematical expressions that describe a part of a whole. They consist of two numbers: a numerator above a line (or slash) and a denominator below. The numerator tells us how many parts we have, while the denominator tells us how many equal parts the whole is divided into.In expressions involving fractions, especially with algebraic terms, understanding these elements becomes crucial. For instance, in handling expressions like \(\left(\frac{2}{t}\right)^{-4}\), after accounting for the negative exponent, we must handle \(\frac{t}{2}\). When it's raised to a power, both the numerator and the denominator are individually raised to that power.Thus, managing fractions well also involves:

• Recognizing reciprocals: Flipping the positions of numerator and denominator.
• Applying powers: Raising both parts of a fraction to a given exponent \(\left(a/b\right)^n = a^n/b^n\).
• Simplifying complex fractions: Making sense of expressions in simpler forms.

Handling fractions with care, particularly in algebraic contexts, empowers problem-solving and expression simplification.

Simplifying Algebraic Expressions

Simplifying expressions is a fundamental part of algebra that involves making an expression easier to work with or to understand. This usually means rewriting it in a more concise and familiar form without changing its value.In the exercise we looked at, we end with the expression \(\frac{t^4}{16}\). Simplification involved these key steps:

• Understanding negative exponents and converting them into positive by flipping the fraction.
• Applying exponents correctly to numerators and denominators separately.
• Finalizing the expression such that all terms are simplified to their lowest terms.

Simplified algebraic expressions are valuable because they reduce complexity and reveal clearer insights into the problem. By mastering these techniques, you develop greater confidence and capability in handling a wide range of algebraic challenges.