Problem 31 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ a^{3} b^{-2} c^{-5} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^{3} b^{-2} c^{-5}\) using only positive exponents. Answer: \(a^{3} \cdot \frac{1}{b^2} \cdot \frac{1}{c^5}\)

Step 1: Identify the negative exponents

In the given expression \(a^{3} b^{-2} c^{-5}\), the negative exponents are for variables \(b\) and \(c\). We need to rewrite these negative exponents as positive exponents.

Step 2: Apply the property of exponents

We will use the property \(a^{-n} = \frac{1}{a^n}\) to rewrite the negative exponents as positive exponents. For variable \(b\), the exponent is -2. Using the property, \(b^{-2} = \frac{1}{b^2}\). For variable \(c\), the exponent is -5. Using the property, \(c^{-5} = \frac{1}{c^5}\).

Step 3: Rewrite the given expression with positive exponents

Now we will substitute the positive exponents for variables \(b\) and \(c\) back into the given expression. \(a^{3} b^{-2} c^{-5} = a^{3} \cdot \frac{1}{b^2} \cdot \frac{1}{c^5}\) The expression with only positive exponents is: $$ a^{3} \cdot \frac{1}{b^2} \cdot \frac{1}{c^5} $$

Key Concepts

Negative Exponents

Understanding negative exponents is critical for working with algebraic expressions. By definition, a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, when we see an expression like \( b^{-2} \), it tells us to take the reciprocal of \( b \) squared, resulting in \( \frac{1}{b^2} \).

Grasping this concept is essential because it enables us to transform expressions with negative exponents into equivalent expressions with only positive exponents, which can be simpler to work with, particularly in more complex equations or when performing operations involving multiple terms with different exponents.

Exponent Properties

Exponent properties, also known as the laws of exponents, are rules that describe how to handle exponents in algebraic operations. For our particular exercise, we employ the property that \( a^{-n} = \frac{1}{a^n} \), which helps move terms with negative exponents to the denominator, thus turning them into positive exponents.

Other important exponent properties include the product of powers property (\( a^m \cdot a^n = a^{m+n} \)), the power of a power property (\( (a^m)^n = a^{mn} \)), and the quotient of powers property (\( \frac{a^m}{a^n} = a^{m-n} \)). Familiarity with these properties enables students to manipulate and simplify expressions methodically.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the given exercise, \( a^{3} b^{-2} c^{-5} \) is an example of an algebraic expression involving three variables raised to different powers.

When dealing with algebraic expressions, it's important to understand how terms are combined and how exponents affect those terms. Terms with the same base can be combined using exponent properties. Expressions can also be transformed to separate or combine terms, making them easier to evaluate or further manipulate.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more concise or readable form without changing their value. The exercise demonstrates simplifying by converting negative exponents into positive ones. The simplification process may involve applying exponent properties, combining like terms, or factoring.

In more complex situations, simplifying an expression might include multiplying out parentheses, dividing polynomials, or rationalizing denominators. The goal is to make the expression as straightforward as possible, ideally to aid in solving equations or enhancing one's understanding of the mathematical relationship between terms in the expression.