Problem 27 - 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. $$ 12 a^{3} b^{2} c^{8}, 12 a^{3} b^{2} c^{8} $$

Step-by-Step Solution

Answer

Answer: The other factor is 1.

Step 1: Identify the given product and factor

We are given the product $$12 a^{3} b^{2} c^{8}$$ and one factor $$12 a^{3} b^{2} c^{8}$$.

Step 2: Calculate the other factor

To find the other factor, we need to divide the product by the given factor: $$ \frac{12 a^{3} b^{2} c^{8}}{12 a^{3} b^{2} c^{8}} $$

Step 3: Simplify the expression

Dividing a number by itself gives 1, and dividing variables with the same exponents results in an exponent of 1: $$ \frac{12}{12} \cdot \frac{a^{3}}{a^{3}} \cdot \frac{b^{2}}{b^{2}} \cdot \frac{c^{8}}{c^{8}} = 1 \cdot a^{0} \cdot b^{0} \cdot c^{0} $$ Since any variable with an exponent of 0 equals 1, the final simplified expression is: $$ 1 \cdot 1 \cdot 1 \cdot 1 = 1 $$ Thus, the other factor is 1.

Key Concepts

Exponents

In algebra, exponents are a way of expressing repeated multiplication of the same number or variable. For example, when we have a term like \(a^{3}\), it's the same as saying \(a \times a \times a\). The base \(a\) is multiplied by itself three times, as indicated by the exponent \(3\). Understanding exponents is crucial because they simplify expressions and make calculations more manageable.

Here’s a quick guide to working with exponents:

• When multiplying two terms with the same base, add their exponents. For example, \(a^{m} \times a^{n} = a^{m+n}\).
• When dividing two terms with the same base, subtract their exponents. For instance, \(a^{m} / a^{n} = a^{m-n}\).
• Any number or variable raised to the power of zero equals one: \(a^{0} = 1\).

Using these properties helps in simplifying expressions and solving equations efficiently. When you understand how exponents work, you can manipulate algebraic expressions with greater ease.

Factorization

Factorization in algebra is the process of breaking down a complex expression into simpler terms, or 'factors', that when multiplied together give the original expression. The original problem of finding another factor comes down to identifying elements that contribute to a product.

Consider the expression \(12 a^{3} b^{2} c^{8}\). Each component of this expression is a factor:

• Number 12 is a factor.
• Variable \(a^{3}\) is a factor which means \(a\) appears three times.
• Variable \(b^{2}\) is a factor, denoting \(b\) appears twice.
• Variable \(c^{8}\) is a factor, indicating \(c\) appears eight times.

This process can also involve recognizing common factors in different expressions. Once common factors are identified, expressions can be simplified or even solved by setting them equal to certain values, such as zero for finding roots. Understanding factorization is key to simplifying and solving complex algebraic expressions.

Simplification

Simplification involves reducing an algebraic expression to its simplest form. This is done by combining like terms, reducing fractions, and employing laws of exponents. Simplifying expressions makes them easier to understand and work with further in algebraic operations.

In our problem, simplifying involves dividing the product by a factor, which turned out to be itself. This means we divide each component:

• \(\frac{12}{12} = 1\), reducing the numerical coefficient.
• For variables, division means cancelling out the same powers or leaving a result where it's 1: \( \frac{a^3}{a^3} = 1, \frac{b^2}{b^2} = 1, \frac{c^8}{c^8} = 1\).

Through this process, we end up with the expression \(1\). The power zero rule (\(a^{0} = 1\)) is utilized here as each variable with equivalent exponents effectively cancels out, leaving us with \(1\).

Mastering simplification helps in evaluating expressions quickly and lays the groundwork for solving equations efficiently.