Problem 75 - Elementary Algebra

Question

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ (-4) a^{6} b^{2} ; a b $$

Step-by-Step Solution

Answer

Question: Find the coefficient of the group of factors \(ab\) in the term \((-4)a^{6}b^{2}\). Answer: \((-4)a^5\)

Step 1: Identify the given term and the group of factors

The given term is \((-4) a^{6} b^{2}\) and the group of factors is \(ab\). Step 2: Divide the given term by the group of factors

Step 2: Divide the given term by the group of factors

Divide the term \((-4) a^{6} b^{2}\) by the group of factors \(ab\). Then we have: $$\frac{(-4) a^{6} b^{2}}{ab}$$ Step 3: Simplify the quotient

Step 3: Simplify the quotient

Using rules of exponents and simplifying the quotient, we have: $$\frac{(-4) a^{6} b^{2}}{ab} = (-4) a^{(6-1)} b^{(2-1)} = (-4) a^{5} b$$ Step 4: Identify the coefficient of the group of factors

Step 4: Identify the coefficient of the group of factors

Now, the given term is in the form \(ab\) multiplied by the coefficient. The coefficient of the group of factors \(ab\) in the simplified term is \((-4)a^5\).

Key Concepts

Understanding Factors

Factors are numbers or expressions that multiply together to get another number or expression. They are like the building blocks of multiplication. In algebra, factors can be variables or constants. For example, in the expression \[ (-4) a^{6} b^{2}, \] both \( a \) and \( b \) are factors, as they can be multiplied by other numbers to get the whole expression.

• Factors help in breaking down expressions into simpler, more manageable parts.
• Identifying factors helps in operations like simplification and solving equations.

Understanding and identifying factors is important because it allows you to see what parts of an expression can be simplified or factored out.

Exploring Exponents

Exponents represent repeated multiplication of the same factor. In the term \( a^6, \)the 6 is the exponent which tells us that \( a \) is multiplied by itself six times. Exponents are a concise way to show large calculations.

• Exponents are written as superscripts in algebraic expressions.
• They follow specific rules such as the power of a product rule, and power of a power rule.

In our example, when dividing \( a^{6} \) by \( a \),we apply the rule of deducting the exponents which leaves us with \( a^{5}. \) This emphasizes the importance of understanding exponent rules for simplification.

The Basics of Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols represent quantities without fixed values, known as variables.

• Algebra uses equations to express relationships.
• It includes the use of operations such as addition, subtraction, multiplication, and division.

When working with expressions like \((-4) a^{6} b^{2}\), algebra becomes crucial as it helps us manipulate and simplify these expressions into understandable parts. The division of terms and simplification using algebraic rules allow us to maintain equality and find the required results more easily.

Achieving Simplification

Simplification involves making an expression easier to work with by reducing it to its most basic form. It is like cleaning up a messy room to only leave the essential items behind. In our problem, the term \((-4) a^{6} b^{2}\) is simplified by dividing out the factors \(ab\).

• Simplification can involve reducing numbers, canceling symbols, or combining like terms.
• It helps in understanding complex expressions by breaking them down into easier parts.

To perform simplification correctly, a good understanding of both factors and exponent rules is necessary. This process not only makes solving equations easier but also helps in identifying important coefficients after reduction.