Problem 59 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{15 a^{8} b^{2}-10 a^{2} b^{5}}{5 a^{3} b^{2}} $$

Step-by-Step Solution

Answer

\( \frac{15 a^{8} b^{2}-10 a^{2} b^{5}}{5 a^{3} b^{2}} = 3a^5 - 2ab^3 \)

Step 1: Break Down Each Term

Identify and write down each term in the numerator separately: Terms = \[ 15a^8b^2 - 10a^2b^5 \].

Step 2: Factor Out Common Factor in Numerator

Factor the numerator by finding the greatest common factor of the terms. The greatest common factor is 5, so factor it out:\[ 5(3a^8b^2 - 2a^2b^5) \].

Step 3: Simplify Each Term

Divide each term inside the parentheses by \( 5a^3b^2 \) separately.For \( 3a^8b^2 / (a^3b^2) = 3a^{8-3} = 3a^5 \).For \( -2a^2b^5 / (a^3b^2) = -2a^{2-3}b^{5-2} = -2ab^3 \).

Step 4: Combine Simplified Terms

Combine the simplified terms from the above step. The division results in: \[ 3a^5 - 2ab^3 \].

Key Concepts

Algebraic Fractions

Algebraic fractions work much like the regular fractions that we're familiar with, but they involve variables along with numbers. An algebraic fraction is simply a ratio of two expressions. Just like with numeric fractions, the goal is often to simplify them, which makes them easier to understand or to use in further calculations. To simplify an algebraic fraction correctly:

• Identify and factor out any common factors in the numerator and the denominator.
• When possible, cancel these out exactly like you would with numbers.

Remember, you can't divide by zero, so any variable that might cause the denominator to be zero is a restriction we must consider. This concept becomes pivotal as you move from simple arithmetic into algebra because it lays the groundwork for understanding more complex rational expressions in algebra.

Greatest Common Factor

The Greatest Common Factor, or GCF, in algebra, is a useful tool that helps in simplifying expressions. It represents the largest factor that two or more terms share. Finding the GCF involves a few simple steps:

• First, break down the numbers and expressions to their prime factors. For expressions involving variables, find the lowest power of each variable that appears in every term.
• Then, multiply these factors together to get the GCF.

For example, if you have terms with numbers like 15 and 10, the GCF is 5, since 5 is the largest number that divides both 15 and 10. In terms of variables like in our exercise with powers of 'a' and 'b', find the smallest exponent for each variable in the expression. Learning to find the GCF is essential, especially when simplifying fractions or factoring polynomials.

Exponent Rules

Exponent rules or laws are guidelines that help us perform operations involving powers of numbers. These rules simplify expressions with exponents, making them easier to calculate and understand. Some important exponent rules include:

• Product Rule: This states that when you multiply like bases, you add their exponents, e.g., \(a^m \cdot a^n = a^{m+n}\).
• Quotient Rule: When dividing like bases, subtract the exponents, e.g., \(a^m / a^n = a^{m-n}\).
• Power Rule: When raising a power to another power, multiply the exponents, e.g., \((a^m)^n = a^{mn}\).

In the step-by-step solution of our exercise, we use the quotient rule extensively. This is crucial for breaking down terms so they can be simplified. By mastering these rules, you'll find it easier to work through problems involving exponents and polynomials.