Problem 12 - Elementary Algebra

Question

For Problems \(1-24\), divide the monomials. $$ \frac{-72 a^{5} b^{4}}{-12 a b^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(6a^{4}b^{2}\).

Step 1: Simplify the Coefficients

Divide the coefficients of the monomials: \(-72 \div -12 = 6\). Hence, the coefficient simplifies to 6.

Step 2: Simplify the Literal Part for 'a'

Apply the quotient rule for exponents for the variable \(a\). Subtract the exponent in the denominator from the exponent in the numerator: \(a^{5} \div a^{1} = a^{5-1} = a^{4}\).

Step 3: Simplify the Literal Part for 'b'

Apply the quotient rule for exponents for the variable \(b\). Subtract the exponent in the denominator from the exponent in the numerator: \(b^{4} \div b^{2} = b^{4-2} = b^{2}\).

Step 4: Combine the Results

Combine the simplified coefficient and literal parts to form the final simplified expression. The result is \(6a^{4}b^{2}\).

Key Concepts

Coefficient Simplification

Simplifying coefficients involves dividing the numbers in front of the variables. In our example, we begin with the coefficients -72 and -12. Looking at what divides both evenly can simplify them. After careful division: -72 divided by -12 results in 6. Since both numbers were negative, dividing them gives a positive result.

• The division part focuses on comparing the fixed numbers, not on the variables or letters themselves.
• Always consider the signs. Two negatives make a positive, while one positive and one negative will result in a negative.

Breaking coefficients down to their smallest forms makes the entire expression easier to handle.

Quotient Rule for Exponents

The quotient rule for exponents is a powerful tool for simplifying expressions with the same base. This rule states that when you divide two powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
For example, if we want to simplify \(a^5 \div a^1\), we follow this process: subtract the exponent 1 from 5, resulting in \(a^{5-1} = a^4\).

• This rule applies independently to each variable involved.
• Remember, the bases need to be identical to apply this rule correctly.

This method allows us to manage expressions with multiple exponent operations effortlessly.

Simplifying Expressions

Combining all simplified parts of a division results in a clear and understandable expression. In our division example, we initially simplified the coefficients to 6. This left us with literal parts where we applied the quotient rule to variables 'a' and 'b'.
By calculating \(a^4\) and \(b^2\), we merged these with the coefficient, leading to the expression 6a^4b^2.

• Always perform one simplification step at a time.
• Merging all parts at the end is essential to obtaining the final simplified outcome.

Taking a step-by-step approach ensures clarity and accuracy throughout the process.