Problem 10 - Technical Mathematics with Calculus

Question

Multiply and reduce. Do some by calculator. $$\frac{15 a^{2}}{7 b^{2}} \cdot \frac{28 a b}{9 a^{3} c}$$

Step-by-Step Solution

Answer

\(\frac{20}{3bc}\)

Step 1: Identify the Numerator and Denominator

Firstly, identify the numerator and denominator in both fractions. The numerator of the first fraction is 15a^2, and the denominator is 7b^2. The numerator of the second fraction is 28ab, and the denominator is 9a^3c.

Step 2: Multiply Numerators and Denominators

Multiply the numerators together and the denominators together. The new numerator is (15a^2)(28ab), and the new denominator is (7b^2)(9a^3c).

Step 3: Simplify the Expression

Before performing the multiplication, cancel any common factors in the numerator and denominator. 15 and 9 can be simplified as they share a factor of 3, and 28 and 7 can be simplified since they share a factor of 7. Furthermore, simplify the variables by canceling common a and b terms. We can reduce the a terms (a^2 and a^3) by subtracting the exponents, and similarly with the b terms.

Step 4: Perform the Multiplication

Now multiply the simplified terms: \[\frac{(15 / 3) \times (28 / 7) \times a^2 \times a \times b}{(7 / 7) \times (9 / 3) \times a^3 \times b^2 \times c}\] which simplifies to \[\frac{5 \times 4 \times a^3 \times b}{3 \times a^3 \times b^2 \times c}\].

Step 5: Final Reduction

Lastly, further reduce the expression by dividing both the numerator and the denominator by a^3 and b which leaves the final result: \[\frac{5 \times 4}{3 \times b \times c} = \frac{20}{3bc}\].

Key Concepts

Simplifying Fractions

When it comes to working with fractions, especially in the multiplication of fractions, simplifying them is a pivotal step. Simplification makes fractions easier to work with and often reveals a more understandable form of an expression. To simplify a fraction, you search for any common factors that the numerator and the denominator share.

For numerical values, this often involves division by common prime factors. For instance, if both the numerator and the denominator are divisible by 3, you can divide both by 3 to simplify the fraction. In algebraic expressions, this simplification extends to variables. You can cancel out the same factor of a variable in both the numerator and the denominator, effectively reducing the powers of the variable by subtracting the exponents, as seen with the terms a^2 and a^3 in the exercise.

• Identify common numerical or variable factors in the numerator and denominator.
• Divide both the numerator and the denominator by those common factors.
• If you're working with variables, subtract the lower exponent from the higher to simplify.

Further simplification might be possible after multiplication, so always check your result for additional simplifications.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, operators, variables, and sometimes exponents, as seen in our multiplication problem. These expressions allow us to represent relationships and formulas that apply to a variety of values. Multiplication of algebraic expressions follows the same rules as multiplying numerical fractions but with the addition of dealing with variables and their exponents.

Handling Variables During Multiplication

When multiplying variables, you add their exponents if the bases are the same. For example, when multiplying a^2 by a, the result is a^(2+1), which equals a^3. If variables have no exponent, they are assumed to have an exponent of 1. It's crucial during multiplication to simplify expressions before performing operations to reduce complexity.

• Multiply coefficients (numerical factors) of the variables in the usual way.
• Add exponents when multiplying like bases.
• Simplify by combining like terms.

Numerical Operations

Numerical operations such as addition, subtraction, multiplication, and division play a fundamental role in mathematics, and the multiplication of fractions is no exception. When multiplying fractions, you multiply the numerators together and the denominators together to get a new fraction.

However, before carrying out the multiplication, it's advantageous to simplify the fractions if possible to avoid working with larger numbers. As in our example, breaking down the numbers 15, 9, 28, and 7 by finding common divisors streamlines the multiplication process. Performing these operations accurately and simplifying wherever possible is essential for a clear and concise solution to mathematical problems involving fractions.

• Multiply numerators together and denominators together after simplifying.
• Always consider further simplification post multiplication to obtain the simplest form.
• Accuracy in performing these operations is key to the reliability of the result.