Problem 76 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{(r t)^{3}}{r t^{4}} \div \frac{\left(r t^{2}\right)^{3}}{r^{2} t^{3}}$$

Step-by-Step Solution

Answer

The final simplified expression is \ \(\frac{r}{t^{4}}\).

Step 1 - Simplify the first fraction

The given first part of the expression is \ \ \ \( \frac{(rt)^{3}}{rt^{4}} \).Rewrite it by raising each factor in the numerator to the power of 3: \((rt)^{3} = r^{3}t^{3}\). So, the expression becomes \ \ \ \( \frac{r^{3}t^{3}}{rt^{4}} \). Now, simplify by canceling out common factors: \( r^{3}/r = r^{2}\) and \( t^{3}/t^{4} = 1/t \). Thus, the simplified first fraction is \ \( \frac{r^{2}}{t} \).

Step 2 - Simplify the second fraction

The second part of the expression is \ \( \frac{(rt^{2})^{3}}{r^{2}t^{3}} \).Rewrite it by raising each factor in the numerator to the power of 3: \((rt^{2})^{3} = r^{3}t^{6}\). So, the expression becomes \ \( \frac{r^{3}t^{6}}{r^{2}t^{3}} \). Now, simplify by canceling out common factors: \( r^{3}/r^{2} = r \) and \( t^{6}/t^{3} = t^{3} \). Thus, the simplified second fraction is \ \( \frac{rt^{3}}{1} = rt^{3} \).

Step 3 - Perform the division of the simplified fractions

Now, divide the simplified expressions from Steps 1 and 2: \ \ \ \( \frac{r^{2}}{t} \div rt^{3} \). Divide by multiplying by the reciprocal of the second fraction: \ \ \ \( \frac{r^{2}}{t} \times \frac{1}{rt^{3}} = \frac{r^{2}}{rt^{4}} \).

Step 4 - Simplify final expression

Simplify \( \frac{r^{2}}{rt^{4}} \) by canceling out common factors: \( r^{2}/r = r \) and \( t^{4} = t^{4} \).\ Therefore, the final simplified expression is \ \ \( \frac{r}{t^{4}} \).

Key Concepts

simplifying fractions

Simplifying fractions means reducing them to their simplest form. You achieve this by dividing both the numerator and the denominator by their greatest common divisor. Here's how it's done:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For example, in the first step of our exercise, we simplified
\( \frac{(rt)^{3}}{rt^{4}} \) to \( \frac{r^{2}}{t} \). This was done by canceling out the common factors in both the numerator and the denominator. Keep in mind that simplifying fractions makes them easier to work with in further calculations.

exponents

Exponents are a way to express repeated multiplication of the same factor. For instance, \( a^{3} \) means \( a \times a \times a \). Some important rules of exponents are:

• \( a^{m} \times a^{n} = a^{m+n} \)
• \( (a^{m})^{n} = a^{mn} \)
• \( \frac{a^{m}}{a^{n}} = a^{m-n} \)

In our example, we applied these rules to simplify the terms:

• \( (rt)^{3} = r^{3} t^{3} \).
• \( (rt^{2})^{3} = r^{3} t^{6} \).

Remembering these fundamental rules will help you manage exponents correctly.

rational expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying rational expressions involves a few steps:

• Factor the numerator and the denominator fully.
• Cancel out the common factors from the numerator and the denominator.

In our exercise, we started with two rational expressions and simplified them step by step: \( \frac{(rt)^{3}}{rt^{4}} \rightarrow \frac{r^{2}}{t} \) and

\( \frac{(rt^{2})^{3}}{r^{2}t^{3}} \rightarrow rt^{3} \). Rational expressions can seem complicated, but breaking them down into smaller parts makes them manageable. Practicing these steps will improve your confidence.

multiplication and division of fractions

Multiplication and division of fractions might seem confusing at first, but they become simple with practice. Here are the key steps:

• To multiply fractions, multiply the numerators together and the denominators together: \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \).
• To divide fractions, multiply the first fraction by the reciprocal (inverse) of the second fraction: \( \frac{a}{b} \rightarrow \frac{d}{c} \).

For instance, in our final step, we divided \( \frac{r^{2}}{t} \) by \( rt^{3} \) by multiplying: \( \frac{r^{2}}{t} \times \frac{1}{rt^{3}} = \frac{r^{2}}{rt^{4}} \). Simplifying this, we get the final expression: \( \frac{r}{t^{4}} \). Mastering these operations enables you to tackle more complex algebraic problems with ease.