Problem 34 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(16 t^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{16}{t^3}\).

Step 1: Rewrite the Expression using Positive Exponents

The expression given is \(16 t^{-3}\). To simplify this expression, we need to transform the negative exponent into a positive exponent. Recall that \(a^{-b} = \frac{1}{a^b}\). Applying this rule to \(t^{-3}\), we get \(t^{-3} = \frac{1}{t^3}\).

Step 2: Combine the Terms

Now that we have transformed the negative exponent to a positive one, rewrite the entire expression. Instead of \(16 t^{-3}\), we have:\[16 \cdot \frac{1}{t^3} = \frac{16}{t^3}\].This is the expression with all positive exponents.

Key Concepts

Negative Exponents

Negative exponents in algebra can often be confusing, but they follow a simple rule that can make them clearer. The core idea is rewriting the expression with a negative exponent to eliminate the negative sign. When you encounter a base with a negative exponent, such as in the expression \(t^{-3}\), you can rewrite it as a fraction: \(t^{-3} = \frac{1}{t^3}\). This transformation from negative to positive exponents is crucial for simplification.

• General Rule: For any non-zero number \(a\) and positive integer \(b\), \(a^{-b} = \frac{1}{a^b}\).
• Example: If you have \(x^{-2}\), rewrite it as \(\frac{1}{x^2}\).
• This rule applies to any base, including constants and variables.

By rewriting the expression, calculations become more straightforward and avoid confusion or mistakes.

Simplifying Expressions

Simplifying expressions is a common task in algebra that makes either numeric or algebraic expressions more manageable by using mathematical rules and techniques. The main goal is to express the equation as simply and clearly as possible, often by removing unnecessary complexity. This involves understanding the properties and operations of exponents.

• Combine Like Terms: Simplifying isn't just about removing negative exponents; it involves combining like terms where possible.
• Consistency in Exponents: If an expression contains a mix of positive and negative exponents, convert them all to positive, as seen in \(16 t^{-3} = \frac{16}{t^3}\).
• Final Form: Always aim to express your answer without negative exponents or fractional exponents unless specifically required.

Through simplification, expressions are transformed into a form that's easier to understand, which aids further operations or solutions.

Positive Exponents

Positive exponents in expressions are considered standard form. They maintain clarity and simplicity, as they do not involve fractions or reciprocal values that negative exponents might introduce. Exponents denote repeated multiplication - for example, \(t^3\) is equivalent to \(t \times t \times t\).

• Simplify Without Fractions: Unlike negative exponents, positive exponents keep the expression straightforward, avoiding fractions.
• Multiplication Implication: A positive exponent indicates how many times a base is used as a factor in multiplication.
• When simplifying expressions like our original problem, \(16 t^{-3}\), transferring to \(\frac{16}{t^3}\) ensures we only use positive exponents, aligning with typical algebraic practice.

Using positive exponents generally culminates with more manageable and comprehensible expressions, supporting further mathematical calculations and insights.