Problem 3 - Elementary Algebra

Question

Write each of the following using exponents. \(2 \cdot 2 \cdot 7 \cdot 7 \cdot 7 \cdot(a-4)(a-4)\)

Step-by-Step Solution

Answer

Question: Rewrite the expression with exponents: \(2 \cdot 7 \cdot 2 \cdot 7 \cdot (a-4) \cdot 7 \cdot (a-4)\) Answer: The given expression can be rewritten using exponents as: \(2^2 \cdot 7^3 \cdot (a-4)^2\)

Step 1: Identify and group the factors

We can see that there are two factors of 2, three factors of 7, and two factors of (a-4). So, let's group them together: \((2 \cdot 2) \cdot (7 \cdot 7 \cdot 7) \cdot ((a-4)(a-4))\)

Step 2: Apply the exponent to each group

Now, we need to apply the exponent rules to each group. An exponent of a product is the product of the same factor repeated several times. So: \(2^2 \cdot 7^3 \cdot (a-4)^2\) The expression using exponents is: \(2^2 \cdot 7^3 \cdot (a-4)^2\)

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols to represent numbers in equations and expressions. It allows us to generalize mathematical operations and solve problems in a systematic way. In our exercise, algebra is used to write numbers in a form that can be simplified using properties like exponents. For instance, instead of writing a multiplication of numbers and expressions, algebra allows us to express them in a more concise form by recognizing patterns and groups.

Factorization

Factorization is the process of breaking down numbers or expressions into their constituent factors that, when multiplied, give the original number or expression. In our exercise, the numbers 2, 7, and the expression \(a-4\) were broken into simpler factors.

• Identifying Factor Groups: We group identical factors together. For example, the number 2 appears twice and the number 7 appears three times as factors.
• Simplifying: Once identified, we can simplify these groups using exponents, which makes handling large expressions easier.

Factorization helps in simplifying expressions to make further calculations or solving equations less cumbersome.

Exponent Rules

Exponents express repeated multiplication of the same factor. For example, instead of multiplying 2 by itself, we write it as \(2^2\) when it's used twice. The exponent is the small number written above and to the right of the base number or expression.
Exponents have several rules that help in simplifying expressions:

• Product of Powers: If we multiply same bases, we add their exponents, \((x^a \cdot x^b = x^{a+b})\).
• Powers of Powers: Raising a power to another power multiplies the exponents, \( (x^a)^b = x^{a \cdot b}\).
• Power of a Product: Applies the exponent to each factor inside parentheses, like in \( (xy)^a = x^a \cdot y^a\).

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, operations, and sometimes exponents that represents a particular value. In our exercise, we started with a raw multiplication expression, which through factorization and application of exponent rules, was transformed into a simpler format using exponents.

• Components: Includes constants (such as the numbers 2 and 7), variables like \(a\), and operations (multiplication, exponentiation).
• Simplification: By rewriting expressions using exponents, we achieve a more manageable form, which helps in solving mathematical problems efficiently.

Understanding the structure of expressions is crucial in algebra, as it allows for manipulation and simplification that are essential to solving equations and inequalities.