Problem 78 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{-2 t}{4 t^{-2}}\right)^{-1} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{t^3}\).

Step 1: Write the expression with positive exponents

The given expression is \( \left( \frac{-2 t}{4 t^{-2}} \right)^{-1} \). First, we rewrite the denominator's negative exponent as a positive exponent by rewriting \( t^{-2} \) as \( \frac{1}{t^2} \). Thus the expression becomes \( \frac{-2t}{4 / t^2} \).

Step 2: Simplify the fraction

The fraction inside the main brackets is \( \frac{-2t imes t^2}{4} \). The \( t imes t^2 \) simplifies to \( t^3 \), resulting in a fraction \( \frac{-2t^3}{4} \).

Step 3: Simplify coefficients

The coefficient \(-2\) divided by \(4\) simplifies to \(-\frac{1}{2}\), so the fraction becomes \(-\frac{1}{2}t^3\).

Step 4: Apply the negative outside exponent

We now address the outer exponent \(-1\), which means we take the reciprocal of the internal expression \(-\frac{1}{2}t^3\). This yields \(-2t^{-3}\).

Step 5: Convert to positive exponent

Finally, convert the expression \(-2t^{-3}\) into positive exponents. Thus, the expression becomes \(-2 \cdot \frac{1}{t^3} = -\frac{2}{t^3}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, which involves reducing expressions to their simplest forms. It's like tidying up your room — making it neat, organized, and easy to understand.

Whenever you're given a complicated expression, your aim is to break it down step-by-step:

• Identify and simplify parts of the expression.
• Combine like terms, such as terms with the same base and exponent.
• Clear out unnecessary parts to make it more straightforward.

For example, simplifying the expression \(-2t^3\) involves seeing that there's nothing left to combine or simplify further once you've simplified exponents and coefficients. Mastering this process is crucial for tackling more complex algebraic problems effectively.

Negative Exponents

Negative exponents can initially seem confusing, but they're a simple concept once you get the hang of it. A negative exponent indicates that the base is on the wrong side of the fraction line, so to speak.

In mathematical terms, \(a^{-n} = \frac{1}{a^n}\). This means you flip the base to the denominator to make the exponent positive. Understanding this simple flip can make working with exponents much easier. For example, in our exercise, we rewrote \(t^{-2}\) as \(\frac{1}{t^2}\), converting the negative exponent into a more manageable form.

Always make sure to convert negative exponents into positive ones in your final answer, as it is standard practice in algebra.

Fraction Simplification

The process of fraction simplification is about making fractions as straightforward as possible. This means reducing the top and bottom numbers by their greatest common factor, whenever possible.

For example, simplifying \(-\frac{2t^3}{4}\) requires dividing both the numerator and the denominator by their common factor, which is 2. This yields \(-\frac{1}{2}t^3\).

Simplifying fractions within algebraic expressions often involves combining this knowledge of numeric simplification with rules of exponents, making the process a blend of different math skills.

Algebraic Expressions

An algebraic expression is an expression built from constants, variables, and the algebraic operations: addition, subtraction, multiplication, division, and powers.

These expressions can look complex, but with practice, they become easier to understand and manipulate. For instance, the expression \(\left(\frac{-2t}{4t^{-2}}\right)^{-1}\) combines many concepts like fractions, negative exponents, and simplification.

When dealing with algebraic expressions, you follow rules of operations and laws of exponents, which help break down and simplify the expression until you're left with a simple and tidy form. Always practice feeding your algebraic expressions through these rules step-by-step, enhancing your problem-solving skills along the way.