Problem 52 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ \frac{3}{10 t^{-3} r^{-1}} $$

Step-by-Step Solution

Answer

The rewritten expression with positive exponents is \(0.3t^3r\)

Step 1: Identify elements with negative exponents

The elements with negative exponents in the given equation have been identified as \(t^{-3}\) and \(r^{-1}\)

Step 2: Apply the rule of exponents

Using the rule \(a^{-n} = \frac{1}{a^n}\), the given equation \(\frac{3}{10t^{-3} r^{-1}}\) transforms to \(\frac{3}{10} \cdot t^3 \cdot r\).

Step 3: Simplify the expression

After multiplication, the simplified expression is \(0.3t^3r\)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of symbols, numbers, and operators that represent mathematical relationships. At their core, these expressions can include constants (specific numbers), variables (symbols like \( t \) and \( r \) that can represent different values), and operations such as addition, subtraction, multiplication, and division. Algebraic expressions can be simple, like \( 2x \), or more complex, like \( \frac{3}{10t^{-3} r^{-1}} \), involving multiple terms and operations.Understanding algebraic expressions is fundamental in algebra, as they form the basis for building and solving equations. Expressions can be manipulated and simplified, which is essential for solving mathematical problems effectively.

Negative Exponents

Negative exponents can often be confusing, but they follow a simple rule: any term with a negative exponent can be rewritten as the reciprocal of the term with a positive exponent. For instance, \( t^{-3} \) can be rewritten as \( \frac{1}{t^3} \).This rule applies to all variables and constants:

• \( a^{-n} = \frac{1}{a^n} \)
• \( x^{-2} = \frac{1}{x^2} \)

In the context of algebraic expressions, switching negative exponents to positive ones helps to simplify expressions and operate with them more efficiently. For example, in the expression \( \frac{3}{10t^{-3} r^{-1}} \), converting negative exponents leads to a friendlier form by essentially flipping the variables with negative powers to the numerator.

Simplification Rule

The simplification of algebraic expressions is a critical skill in algebra, helping to produce the most reduced form of an expression. Simplification involves carrying out operations and applying rules, such as the exponent rules, to make the expression as simple as possible. This process often involves:

• Combining like terms
• Performing arithmetic operations
• Applying exponent rules, such as \( a^{-n} = \frac{1}{a^n} \)

In our example, the expression \( \frac{3}{10t^{-3} r^{-1}} \) is simplified by recognizing and dealing with the negative exponents first, transforming it into \( \frac{3}{10} \cdot t^3 \cdot r \).After converting to a multiplication expression, evaluating the constants can further simplify it: \( 0.3t^3r \). This simple form makes it easier to work with mathematically and prepare for further operations or evaluations.