Problem 59 - Algebra 1: Concepts and Skills

Question

Simplify. $$ 18 c^{3} \div \frac{-27 c}{-4} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(72c^{2}\)

Step 1: Transform the Division into Multiplication

Transform the division operation into a multiplication operation by applying the following rule: \(a \div \frac{b}{c} = a * \frac{c}{b}\). This will result in: \(18c^{3} * \frac{-4}{-27c}\)

Step 2: Perform Multiplication Operation

Firstly multiple coefficients i.e. numbers i.e. 18 and -4 which will give -72. Secondly multiply variables i.e. \(c^{3}\) and \(c^{-1}\) where \(c^{-1}\) is inverse of c. Using the rule \(a^{m}*a^{n}=a^{m+n}\), we get \(c^{3-1}=c^{2}\). Putting these together, we get \(-72c^{2}\)

Step 3: Simplify the Sign of the Result

Since we have negative times negative which equals to positive so the final result will be \(72c^{2}\)

Key Concepts

Division of Polynomials

Understanding the division of polynomials is essential when simplifying algebraic expressions involving complex terms. Just like numerical division, polynomial division involves distributing the divisor into the dividend. However, with algebraic expressions, we must pay attention to the variables and their exponents as well.

When dividing terms within a polynomial, it's crucial to invert the term by which we're dividing. This converts a division problem into a multiplication problem, which is often easier to compute. For example, the expression \( 18c^{3} \div \frac{-27 c}{-4} \) becomes \( 18c^{3} * \frac{-4}{-27c} \) when applying this principle. From here, we simply multiply the coefficients, which are the numeric parts, and then apply the laws of exponents to the variables.

The key takeaway in polynomial division is that we're still following the same rules of arithmetic but applied to algebraic terms. This process becomes intuitive with practice and when we remember to flip the divisor and change the operation from division to multiplication.

Multiplication of Algebraic Terms

The multiplication of algebraic terms involves combining the coefficients and then dealing with the variables separately. This is typically straightforward when the bases (variables) are the same and only the exponents differ.

In the given example, we multiply \(18 * -4 \) resulting in \( -72 \). When multiplying variables, we employ the rule \(a^{m}*a^{n} = a^{m+n}\), which simply means we add the exponents of like bases. In our case, \(c^{3} * c^{-1} = c^{3-1} = c^{2}\), because we add the exponents 3 and -1 together. Thus, \( -72c^{2} \) is the product of the multiplication.

Tip for Success

Remember to always multiply coefficients with coefficients and variables with variables. Combine them at the end for the final simplified form.

Negative Exponents

Negative exponents in algebraic expressions can be confusing at first, but they follow a consistent rule that makes them manageable. The rule is that \( a^{-n} = \frac{1}{a^n} \), which means a term with a negative exponent is equal to the reciprocal of that term with a positive exponent.

In the context of the current problem, we notice that when we transformed the division into multiplication, it effectively changed the exponent of c from 1 to -1, resulting in \( c^{-1} \). When we multiply \( c^{3} \) by \( c^{-1} \) we simply subtract the exponents, per the rules of exponents \( (c^{m} * c^{n} = c^{m-n}) \) giving us \( c^{2} \). It is crucial to remember that a negative exponent does not mean the term is negative; it merely indicates the reciprocal.

Through repeated practice and application of the rule for negative exponents, students will find that what once seemed a complex concept becomes second nature, allowing them to simplify expressions with confidence.