Problem 8 - Beginning and Intermediate Algebra

Question

Multiply. $$\left(-\frac{8}{9} c^{5}\right)\left(\frac{3}{10} c^{7}\right)$$

Step-by-Step Solution

Answer

\(-\frac{4}{15} c^{12}\)

Step 1: Multiply the coefficients

First, multiply the coefficients (-8/9 and 3/10) together: \[-\frac{8}{9} \cdot \frac{3}{10}\]

Step 2: Combine the c terms

Next, consider the variable c with its exponents: \(c^5 \cdot c^7\). To multiply these together, add the exponents (since the base c is the same): \[c^{5+7}\]

Step 3: Calculate the final solution

Now, combine the calculations from steps 1 and 2 to get the final solution: \[-\frac{8}{9} \cdot \frac{3}{10} \cdot c^{5+7}\]

Step 4: Simplify the expression

Simplify the expression to obtain the final result: \[-\frac{24}{90} c^{12}\]

Step 5: Reduce the fraction

Reduce the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 6: \[-\frac{4}{15} c^{12}\] In conclusion, the result of multiplying \[\left(-\frac{8}{9} c^{5}\right)\left(\frac{3}{10} c^{7}\right)\] is \[-\frac{4}{15} c^{12}\].

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process that involves some easy steps. Typically, when you multiply two fractions, you multiply the numerators together and then the denominators. For the example at hand, you have two fractions: \(-\frac{8}{9}\) and \(\frac{3}{10}\). Here's what happens:

• Multiply the numerators: \(-8 \cdot 3 = -24\)
• Multiply the denominators: \(9 \cdot 10 = 90\)

So, the product of the fractions is \(-\frac{24}{90}\). It's essential to understand that multiplying fractions does not require finding a common denominator as in addition or subtraction of fractions. You simply proceed with multiplying straight across the top and the bottom.

Exponent Rules

Understanding exponent rules is key when dealing with variables that have powers, such as \(c^5\) and \(c^7\). The power of a variable signifies how many times the base is used as a factor in multiplication. When multiplying terms that have the same base (in this case, the base is \(c\)), you add the exponents. This is because these rules of exponents state:

• \(a^m \cdot a^n = a^{m+n}\)

In the example, multiplying \(c^5\) and \(c^7\) results in \(c^{5+7}\), which simplifies to \(c^{12}\). This rule makes it easy to handle powers of the same base by directly adding their exponents together.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. Once you've multiplied the coefficients and added the exponents, as in \(-\frac{24}{90} c^{12}\), you should simplify to make the expression easier to understand. Simplifying involves:

• Reducing fractions to their lowest terms.
• Ensuring variables are expressed with the simplest exponents when applicable.

The original result \(-\frac{24}{90} c^{12}\) simplifies because the goal is to make the fraction as compact as possible without changing its value. This leads us to find the Greatest Common Divisor of the numerator and denominator.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is crucial when simplifying fractions, as it helps to reduce the fraction to its simplest form. To do this, identify the largest number that divides both the numerator and the denominator without leaving a remainder. In the fraction \(-\frac{24}{90}\), both 24 and 90 can be divided by 6, which is their GCD.

• Divide 24 by 6, getting -4.
• Divide 90 by 6, getting 15.

Thus, \(-\frac{24}{90}\) simplifies to \(-\frac{4}{15}\). With the fraction reduced, the expression \(-\frac{4}{15} c^{12}\) is now in its simplest form. Using the GCD ensures that the result is both accurate and easy to interpret.