Problem 68 - 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. $$ \frac{t^{-3}}{2 t^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{2t^2} \).

Step 1: Rewrite Using Positive Exponents

When dealing with negative exponents, remember that \( a^{-n} = \frac{1}{a^n} \). Therefore, for the expression \( \frac{t^{-3}}{2 t^{-1}} \), we can rewrite it as \( \frac{1}{t^{3}} \) on the numerator and \( \frac{1}{2t} \) on the denominator.

Step 2: Combine the Fractions

The expression with separate fractions is \( \frac{1/t^3}{2/t} \). Dividing by a fraction is the same as multiplying by its reciprocal, so we rewrite it as \( 1/t^3 \times t/2 \).

Step 3: Simplify the Expression

We can simplify the expression \( \frac{t}{2t^3} \) by cancelling one \( t \) in the numerator and denominator, resulting in \( \frac{1}{2t^2} \).

Step 4: Write the Result Using Positive Exponents

The simplified expression is \( \frac{1}{2t^2} \), which already uses positive exponents, so no further changes are necessary.

Key Concepts

Negative Exponents

Negative exponents play a crucial role in mathematics, especially in simplifying expressions. When you see a negative exponent like \( a^{-n}\), it means you're dealing with the reciprocal of the base raised to a positive exponent: \( \frac{1}{a^n}\).

In simpler terms:

• \( x^{-2} \) is the same as \( \frac{1}{x^2} \).
• \( y^{-3} \) turns into \( \frac{1}{y^3} \).

This conversion helps in transforming complex expressions into more manageable ones. To solve expressions with negative exponents, switch them to positive by taking reciprocals.

This transformation enables you to see the expression in a simplified form, aiding further calculations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest possible form. This often requires applying the rules of exponents to combine like terms or eliminate negative exponents.

In this exercise, after converting negative exponents to positive, you're left with a simple algebraic fraction. For example, given \( \frac{1/t^3}{2/t} \), the simplification process requires shifting from division to multiplication by the reciprocal:

• Convert \( \frac{1}{t^3} \) and \( \frac{2}{t} \) into a single fraction by multiplying.
• The result is an expression like \( \frac{t}{2t^3} \).

The next step is to cancel out like terms. Since both parts have a common term \( t \), you can reduce the expression further by canceling:

The final, simplified form is \( \frac{1}{2t^2} \). This process highlights key algebraic manipulation skills.

Positive Exponents

When working with expressions, having positive exponents generally makes them easier to handle. Positive exponents are straightforward and don't involve reciprocals as negatives do.

For example, the exercise we reviewed converts the expression with negative exponents into one with positive exponents:

• The original expression was \( \frac{t^{-3}}{2t^{-1}} \).
• Through conversion and simplification, it ends up as \( \frac{1}{2t^2} \).

This final result with positive exponents allows you to more easily interpret and work with the expression in subsequent calculations or applied contexts.

Positive exponents essentially keep the expression in its simplest form, especially if further operations are needed. The vast majority of mathematical interactions prefer expressions in this clean, clear form.