Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \left(\frac{r^{n}}{r^{5-2 n}}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression, using positive exponents only, is: $$ r^{12n-20} $$

Step 1: Distribute the exponent to both numerator and denominator

Raise both the numerator and the denominator of the fraction to the power of 4: $$ \left(\frac{r^{n}}{r^{5-2n}}\right)^{4} = \frac{(r^{n})^4}{(r^{5-2n})^4} $$

Step 2: Apply the power rule to the exponents

Using the power rule, we multiply the exponents by 4: $$ \frac{(r^{n})^4}{(r^{5-2n})^4} = \frac{r^{4n}}{r^{4(5-2n)}} $$

Step 3: Distribute the exponent in the denominator

Distribute the exponent 4 in the denominator: $$ \frac{r^{4n}}{r^{4(5-2n)}} = \frac{r^{4n}}{r^{20-8n}} $$

Step 4: Use the quotient rule for exponents

The quotient rule states that when dividing expressions with the same base and different exponents, we subtract the exponent in the denominator from the exponent in the numerator: $$ \frac{r^{4n}}{r^{20-8n}} = r^{(4n)-(20-8n)} $$

Step 5: Simplify the expression

Now, simplify the expression by performing the subtraction of exponents: $$ r^{(4n)-(20-8n)} = r^{4n-20+8n} = r^{12n-20} $$

Step 6: Write the final expression

The simplified expression, using positive exponents only, is: $$ r^{12n-20} $$

Key Concepts

Operations with Exponents

When working with exponents, also known as powers, there are certain rules and operations that make calculations much easier. Exponents indicate how many times a number, the base, is multiplied by itself. For example, the expression 2^3 means 2 multiplied by itself 3 times, resulting in 8.

Operations with exponents become more complex when they involve variables, such as in the expression r^n, which means r multiplied by itself n times, where n is an exponent. The operation also becomes interesting when we deal with a fraction of exponents, where both the numerator and denominator have exponents, as in the exercise (r^n)/(r^{5-2n}).

There are several rules for handling such operations effectively: the product rule, quotient rule, power rule, and rules for dealing with negative exponents. By applying these rules step by step, we can simplify complex exponential expressions to arrive at a much simpler equivalent form.

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental law used when dividing exponential expressions with the same base. According to this rule, when dividing expressions like a^m / a^n, we subtract the exponent in the denominator from that in the numerator, thus a^m / a^n = a^{m-n}.

This rule makes it easy to reduce expressions without performing long, tedious multiplications. For example, if we divide r^4 by r^2, we would subtract 2 from 4 to end up with r^(4-2) = r^2. In the provided exercise, we have the fraction r^{4n}/r^{20-8n}, and we use the quotient rule to merge the exponents into a single term, leading to a simpler expression r^{(4n)-(20-8n)}.

Power Rule for Exponents

Exponential expressions can get even trickier when we deal with powers of powers, like (a^m)^n. For such cases, the power rule for exponents comes to the rescue. This rule states that when raising an exponent to another exponent, we multiply the two exponents together. So the expression (a^m)^n becomes a^{mn}.

In our exercise, applying the power rule transforms (r^n)^4 to r^{4n} and (r^{5-2n})^4 to r^{4(5-2n)}, providing a clear path to simplification. Understanding and correctly applying the power rule is crucial for manipulating exponential expressions efficiently and effectively.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying a combination of the rules for operations with exponents to rewrite them in their simplest form. This process typically involves reducing the expression to a base raised to a single exponent, whenever possible.

Let's review our exercise: starting with ((r^n)/(r^{5-2n}))^4, after applying the power and quotient rules, we simplify the subtraction 4n-(20-8n) to eventually arrive at the expression r^{12n-20}, demonstrating a clear, manageable result.

The key to simplifying exponential expressions is to apply the rules in a step-by-step manner and perform any algebraic simplifications such as adding or subtracting exponents systematically. This process allows us to attain an expression that is succinct and often easier to work with in further calculations or applications.