Problem 18 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{27 a^{7} b^{4}}{39 b} \cdot \frac{13 a^{4} b^{2}}{16 a^{5}} $$

Step-by-Step Solution

Answer

Question: Multiply the two fractions and simplify the result: $$ \frac{27 a^{7} b^{4}}{39 b} \cdot \frac{13 a^{4} b^{2}}{16 a^{5}} $$ Answer: The simplified expression after multiplying and simplifying the result is: $$ \frac{13 a^{6} b^{5}}{24} $$

Step 1: Multiply the numerators together

To start, multiply the numerators of the two fractions together: $$ (27 a^{7} b^{4}) \cdot (13 a^{4} b^{2}) $$ Multiply the corresponding constants and combine the terms with the same variable using the properties of exponents: $$ (27 \cdot 13) (a^{7} \cdot a^{4}) (b^{4} \cdot b^{2}) \\ 351 a^{11} b^{6} $$

Step 2: Multiply the denominators together

Next, multiply the denominators of the two fractions together: $$ (39 b) \cdot (16 a^{5}) $$ Multiply the corresponding constants and combine the terms: $$ (39 \cdot 16) a^{5} b \\ 624 a^{5} b $$

Step 3: Combine the multiplied numerators and denominators

Now, combine the result from step 1 with the result from step 2: $$ \frac{351 a^{11} b^{6}}{624 a^{5} b} $$

Step 4: Simplify the resulting fraction

Finally, simplify the fraction by canceling out the common factors. Divide both the numerator and denominator by the greatest common divisor (27) and apply the properties of exponents: $$ \frac{351 a^{11} b^{6}}{624 a^{5} b} = \frac{13 a^{(11-5)} b^{(6-1)}}{24 b^{(1-1)}} \\ \frac{13 a^{6} b^{5}}{24} $$ So, the simplified expression is: $$ \frac{13 a^{6} b^{5}}{24} $$

Key Concepts

Multiplying Algebraic Fractions

Understanding how to multiply algebraic fractions is a fundamental skill in algebra many students must master. Just like multiplying numerical fractions, when multiplying algebraic fractions, you multiply the numerators (the top parts) together and multiply the denominators (the bottom parts) together.

The process to multiply algebraic fractions involves several steps. Let's break down these steps:

• First, identify the numerators and denominators of the fractions you plan to multiply.
• Next, perform the multiplication of the numerators. This often involves multiplying constants (numbers) as well as combining like terms—those having the same variables with the same exponents. During this step, use the properties of exponents to simplify terms like \( a^m \cdot a^n = a^{m+n} \).
• After multiplying the numerators, do the same with the denominators. Multiply any constants together and apply the properties of exponents to variables.
• Once you have your new numerator and denominator, write out the new algebraic fraction.
• Last, simplify the fraction if possible by canceling out any common factors between the numerator and denominator.

By following these steps, you can multiply algebraic fractions to simplify expressions and solve more complex equations.

Properties of Exponents

Properties of exponents, also known as the laws of exponents, are rules that describe how to handle exponents in mathematical expressions. These properties are essential when working with algebraic fractions, especially during multiplication and simplification processes.

Here are the key properties of exponents often used in algebra:

• Product of Powers: When multiplying two expressions with the same base, add the exponents. This is represented as \( a^m \cdot a^n = a^{m+n} \).
• Quotient of Powers: When dividing two expressions with the same base, subtract the exponents. In terms of algebraic fractions, \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When taking a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Zero Exponent: Any base with an exponent of zero equals 1: \( a^0 = 1 \).
• Negative Exponent: A negative exponent represents the reciprocal of the base with a positive exponent: \( a^{-n} = \frac{1}{a^n} \).

With a solid grasp of these properties, you can simplify algebraic expressions efficiently, making it easier to work through equations and arithmetic involving variables and exponents.

Simplifying Rational Expressions

Simplifying rational expressions is the process of reducing an algebraic fraction to its simplest form. This can involve canceling common factors in the numerator and the denominator, as well as applying properties of exponents to simplify variables.

Here's how you simplify a rational expression:

• Factor both the numerator and the denominator into their prime factors if they are numerical, or factor out common algebraic terms.
• Identify any common factors or terms between the numerator and the denominator.
• Divide out the common terms from both the numerator and the denominator to cancel them.
• Take care to respect the properties of exponents when simplifying terms with variables. Reduce expressions like \( a^{m}/a^{n} \) using the Quotient of Powers property to \( a^{m-n} \).
• If the expression has coefficients (numerical factors), reduce them to their lowest terms by dividing by their greatest common divisor.

Simplifying rational expressions not only makes them easier to work with and understand, but it is also a necessary skill for solving equations and working with algebraic functions. While the process can sometimes be complex, systematically factoring and canceling terms will lead to a more manageable expression.