Problem 22 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{-25 m^{4} n^{3}}{14 r^{3} s^{3}} \cdot \frac{21 r s^{4}}{10 m n} $$

Step-by-Step Solution

Answer

Answer: $$-\frac{63}{8} m^3 n^2 r^{-2} s^1$$

Step 1: 1. Identify the variables and coefficients

In the given expression, we have two fractions, and each fraction has a numerator and a denominator. First, we need to identify the variables and their coefficients: $$ \frac{-25 m^{4} n^{3}}{14 r^{3} s^{3}} \cdot \frac{21 r s^{4}}{10 m n} $$

Step 2: 2. Multiply the numerators together

Next, we will multiply the numerators of both fractions together. In this case, we multiply the coefficients and combine like terms: $$ (-25 m^{4} n^{3}) \cdot (21 r s^{4}) = -525 m^{4} n^{3} r s^{4} $$

Step 3: 3. Multiply the denominators together

Similarly, we will multiply the denominators of both fractions together: $$ (14 r^{3} s^{3}) \cdot (10 m n) = 140 m n r^{3} s^{3} $$

Step 4: 4. Combine the numerators and denominators

Now that we have the product of the numerators and denominators, let's write the resulting fraction: $$ \frac{-525 m^{4} n^{3} r s^{4}}{140 m n r^{3} s^{3}} $$

Step 5: 5. Simplify the fraction

Finally, we'll simplify the expression by canceling out any common factors between the numerator and denominator. We can see that both the numerator and denominator have common factors of 5, m, n, r, and s: $$ \frac{-525}{140} = \frac{-21 \cdot 5^1 \cdot 3^1 \cdot 7^1}{2^3 \cdot 5^1 \cdot 7^1} = \frac{-21 \cdot 3}{2^3} = -\frac{63}{8} $$ As for the variables, simplifying gives: $$ \frac{m^{4} n^{3} r s^{4}}{m n r^{3} s^{3}} = m^{(4-1)} n^{(3-1)} r^{(1-3)} s^{(4-3)} = m^3 n^2 r^{-2} s^1 $$

Step 6: 6. Combine the results

Finally, combining the simplified coefficients and the variables, we get: $$ -\frac{63}{8} m^3 n^2 r^{-2} s^1 $$

Step 7: 7. Write the final answer

The result of the multiplication and division is: $$ \frac{-25 m^{4} n^{3}}{14 r^{3} s^{3}} \cdot \frac{21 r s^{4}}{10 m n} = -\frac{63}{8} m^3 n^2 r^{-2} s^1 $$

Key Concepts

Fraction Multiplication

Fraction multiplication is straightforward! You take two fractions, multiply the numerators (the top numbers), and then multiply the denominators (the bottom numbers). This results in a new fraction containing both products. In the exercise, we had two fractions that needed multiplying:

• First fraction: \( \frac{-25 m^{4} n^{3}}{14 r^{3} s^{3}} \)
• Second fraction: \( \frac{21 r s^{4}}{10 m n} \)

To multiply them, the numerators \(-25 m^4 n^3\) and \(21 r s^4\) were multiplied, creating a new numerator: \(-525 m^4 n^3 r s^4\). Likewise, the denominators \(14 r^3 s^3\) and \(10 m n\) were multiplied, producing the denominator: \(140 m n r^3 s^3\). Simply put, fraction multiplication allows us to combine or increase expressions using multiplication.

Fraction Division

Fraction division works by flipping, or finding the reciprocal of the second fraction, and then multiplying. The reciprocal of a fraction is essentially swapping its numerator and denominator. In our problem, however, we didn't have a direct division of fractions; rather, we used multiplication rules. However, if you had: \[\left(\frac{a}{b}\right) \div \left(\frac{c}{d}\right)\]You would rearrange this as multiplying the first by the reciprocal of the second:\[\frac{a}{b} \cdot \frac{d}{c}\]Hence, fraction division relies on the multiplication process with an added pre-step of flipping the secondary fraction.

Simplifying Expressions

Simplifying is all about making expressions easier to handle. After multiplying the numerators and denominators in our exercise, simplification became necessary to reduce the expression to its smallest form. The immediate task here was to identify and cancel common factors in both the numerator and the denominator. For the given problem:

• The common numerical factor was 5, which reduced \(\frac{-525}{140}\) to \(-\frac{63}{8}\)
• Common variable factors such as \(m, n, r, \text{and} s\) were present in both parts, simplifying further.

By canceling these out, the expression became less cluttered: \( m^3 n^2 r^{-2} s\). Thus, simplifying aids in clearer representation, removing unnecessary complexity.

Variable Exponents

When working with expressions, handling variable exponents correctly is crucial. In expressions like ours, where multiplication and division of variables occur, exponents determine the power to which the variables are raised. Take the expression \(m^4\) divided by \(m^1\). Here, subtract the exponents (i.e., \(4 - 1 = 3\)) to get \(m^3\). Similarly for other variables:

• \(n^3/n^1 = n^{2}\)
• \(r^1/r^3 = r^{-2}\)
• \(s^4/s^3 = s^1\)

Exponents simplify expressions by combining variables with the same base, either multiplying by adding exponents or dividing by subtracting them. It's a useful tool for managing large numbers and complex expressions effectively.