Problem 37 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ a^{2} b^{3} c^{-2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the negative exponent

In the given expression \(a^2 b^3 c^{-2}\), the negative exponent is -2 for the term \(c^{-2}\).

Step 2: Apply the property to the negative exponent term

Since \(c^{-2}\) has a negative exponent, rewrite it using the property \(a^{-n} = \frac{1}{a^n}\). Therefore, \(c^{-2} = \frac{1}{c^2}\).

Step 3: Substitute the rewritten term in the original expression

Replace the term \(c^{-2}\) with \(\frac{1}{c^2}\) in the original expression. So, the expression becomes: $$a^2 b^3 c^{-2} = a^2 b^3 \frac{1}{c^2}$$.

Step 4: Simplify the expression

As there are no more simplifications to be done, the final expression with only positive exponents is: $$a^2 b^3 \frac{1}{c^2}$$.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. They are a core part of algebra used to describe mathematical relationships and can represent real-world scenarios. Each component in an algebraic expression is an important piece of the puzzle. Here's a closer look:

• **Variables**: These are symbols (like \(a, b, c\)) that represent unknown values. In most cases, letters from the alphabet are used as variables.
• **Constants**: Fixed numbers or values added to or subtracted from variables in the expression.
• **Coefficients**: Numbers multiplied by the variables. For example, in \(3x\), 3 is the coefficient.
• **Operators**: Mathematical symbols like \(+, -, \times, \)** that indicate the operation to be performed between terms.

Algebraic expressions are the building blocks of equations and inequalities. They allow us to describe sequences, series, and various relationships with a clean concise structure. Transforming them into different forms, like rewriting expressions with only positive exponents, can simplify many mathematical operations.

Negative Exponents

Negative exponents are a concept that may initially seem complex, but they carry a simple meaning. A negative exponent indicates that the base of that power needs to be taken as a reciprocal. The concept is best explained with an example:

• The expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).

This means that rather than multiplying the base several times, as with positive exponents, we divide 1 by the base raised to the positive of that exponent. The use of negative exponents is very useful:

• It allows us to simplify expressions and solve algebraic equations more efficiently.
• It helps in converting products into quotients, making division operations more manageable.

Shifting to positive exponents, as demonstrated in the given exercise, not only makes it easier to interpret an algebraic expression, but also aligns better with most standard mathematical practices.

Properties of Exponents

Understanding the properties of exponents is crucial for manipulating and simplifying expressions effectively. These properties allow us to rewrite expressions in a more usable form. Here are some essential properties related to exponents:

• **Product of Powers Property**: If you multiply two expressions with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers Property**: If you divide two expressions with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• **Power of a Power Property**: If you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).
• **Negative Exponent Property**: Converts a negative exponent into a positive one by taking the reciprocal: \(a^{-n} = \frac{1}{a^n}\).

Applying these properties is what allows us to revise expressions and solve mathematical problems with a clearer strategy, as showcased in the original problem. They are especially useful in calculus, algebra, and geometry, where changes in expressions influence the broader scope of the problem-solving process.