Problem 22 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ -121 a^{6} b^{8} c^{10}, 11 b^{2} c^{5} $$

Step-by-Step Solution

Answer

Question: Given the product \(-121a^{6}b^{8}c^{10}\) and the factor \(11b^{2}c^{5}\), find the other factor. Answer: The other factor is \(-11a^{6}b^{6}c^{5}\).

Step 1: Identify the given product and factor

The given product is \(-121a^{6}b^{8}c^{10}\), and one of its factors is \(11b^{2}c^{5}\).

Step 2: Divide the given product by the factor

Divide the given product by the given factor using the rules for exponents: $$ \frac{-121a^{6}b^{8}c^{10}}{11b^{2}c^{5}} $$

Step 3: Simplify the division

Simplify the division by dividing numerical coefficients and subtracting exponents of corresponding variables: $$ \frac{-121}{11} \cdot a^{(6-0)} \cdot b^{(8-2)} \cdot c^{(10-5)} = -11a^{6}b^{6}c^{5} $$

Step 4: Write the other factor

The other factor is \(-11a^{6}b^{6}c^{5}\).

Key Concepts

Factors

In algebra, factors are crucial when dealing with expressions and equations. Simply put, a factor is a number or expression that can be multiplied by another, to produce a given product. Let's think about the numbers first. For example, in the product of 15, 3 and 5 are factors because 3 multiplied by 5 gives you 15.
In algebraic expressions, factors usually include variables with exponents. For example, consider the expression \(a^2b\). Here, both \(a^2\) and \(b\) are factors of the expression. The understanding of factors extends to expressions with coefficients and powers of variables.
When presented with a product and a known factor, the goal is to find the missing factor that pairs with the given one to create the complete product. It involves dividing the product by the known factor, which leads us to simplification.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. In the expression \(a^6\), "6" is the exponent and "a" is the base, implying \(a\) is multiplied by itself six times. Exponents are particularly useful in expressing large quantities compactly.
In algebraic manipulation, knowing how to handle exponents is key. During division, you subtract the exponents when the bases are the same. For example, dividing \(b^8\) by \(b^2\) simplifies to \(b^{(8-2)} = b^6\). This property helps to simplify expressions and helps to find missing factors effectively.

• When multiplying like bases, add the exponents.
• When dividing like bases, subtract the exponents.
• The power of a power means multiplying the exponents.

By remembering these principles, navigating exponential expressions becomes simpler.

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. This entails combining like terms and making expressions easier to understand and work with. Simplification is a crucial tool when solving problems or finding a missing factor, as it reduces the complexity of the expression.
To simplify, you must carefully follow the rules of arithmetic operations, including handling coefficients properly and applying the laws of exponents to the variable components. Let's look at our problem:

• Start by addressing the coefficients: Divide \(-121\) by \(11\) to get \(-11\).


• Apply the exponent rules: Subtract the exponents for each variable with the same base like \(b^{8-2}\) and \(c^{10-5}\).


• Combine the results to find the simplest form of the expression.

These steps ensure that you're left with the simplest possible term, making your calculations straightforward and your end results clear.