Problem 19 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \left[\frac{r^{6} s^{-2}}{m^{-5} n^{4}}\right]^{-4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\left[ \frac{r^6 s^{-2}}{m^{-5} n^4} \right]^{-4}$$ with only positive exponents. Answer: $$\frac{s^8 n^{16}}{r^{24} m^{20}}$$

Step 1: Apply the exponent to all terms inside brackets

According to the first property of exponents, we have: $$ \left[ \frac{r^6 s^{-2}}{m^{-5} n^4} \right]^{-4} = \frac{(r^6)^{-4} (s^{-2})^{-4}}{(m^{-5})^{-4} (n^4)^{-4}} $$

Step 2: Simplify the exponents according to exponent laws

Now, we simplify the exponents in each term using the property $$a^{m.n} = a^{m\times n}$$: $$ \frac{(r^6)^{-4} (s^{-2})^{-4}}{(m^{-5})^{-4} (n^4)^{-4}} = \frac{r^{-24} s^8}{m^{20} n^{-16}} $$

Step 3: Eliminate negative exponents using properties of exponents

Use the property $$a^{-n} = \frac{1}{a^n}$$ to eliminate the negative exponents: $$ \frac{r^{-24} s^8}{m^{20} n^{-16}} = \frac{r^{-24} s^8}{m^{20}} \cdot \frac{1}{n^{-16}} =\frac{s^8}{r^{24} m^{20}} \cdot n^{16} $$

Step 4: Combine the terms

Finally, we combine the terms to get the simplified expression: $$ \frac{s^8}{r^{24} m^{20}} \cdot n^{16} = \frac{s^8 n^{16}}{r^{24} m^{20}} $$ Therefore, the simplified expression with only positive exponents is: $$ \frac{s^8 n^{16}}{r^{24} m^{20}} $$

Key Concepts

Exponent Laws

Understanding exponent laws is crucial when simplifying expressions, especially those involving various power terms. Exponent laws, also known as exponent rules, are a set of properties that govern the operations on numbers with exponents. In the exercise presented, two important laws are used. The Power of a Power rule, \(a^{m})^{n} = a^{m\times n}\), allows us to multiply exponents when a term is raised to another power. Additionally, there's the Negative Exponent rule, which states that \(a^{-n} = \frac{1}{a^{n}}\), making it possible to turn a negative exponent into a positive one by placing the term in the denominator. Applying these rules systematically, as seen in the exercise, helps to simplify complex algebraic expressions with exponents into a more digestible format.

Negative Exponents

Negative exponents can often cause confusion, but they follow a simple principle. The negative sign indicates the reciprocal of the base raised to the positive exponent. In simpler terms, any number with a negative exponent, such as \(a^{-n}\), can be rewritten as \(\frac{1}{a^{n}}\). This principle is part of the properties of exponents and plays a pivotal role in transforming algebraic expressions with negative exponents into an equivalent form that only has positive exponents, as required in many mathematical contexts, including the provided exercise. By understanding this property, simplifying terms becomes a more straightforward process.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. The complexity of these expressions can range from simple, such as \(3x + 2\), to more complicated forms involving exponents and multiple variables, like the one in our example. A key aspect of understanding algebraic expressions is recognizing how to apply the properties of exponentiation to simplify the expression. This involves not only knowing the exponent rules but also being comfortable with the manipulation of the variables and constants. The exercise illustrates how to handle both negative exponents and multiple variables in an algebraic expression to arrive at a simplified form.

Properties of Exponents

The properties of exponents are the backbone of any operation that deals with powers. These properties include the Power of a Product, Power of a Quotient, Power of a Power, and rules for dealing with Zero and Negative Exponents. In our exercise, after applying the first property to distribute the exponent across all terms inside the brackets, we then utilized the Power of a Power and the Negative Exponent properties to simplify the expression. Through these steps, the complex expression with negative and positive exponents is simplified so no negative exponents remain. These properties are essential tools for students to master in order to successfully work with exponential expressions in algebra.