Problem 19 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. \(a\) cubed minus \((b+7)\) squared

Step-by-Step Solution

Answer

Question: Rewrite the expression \(a^3 - (b+7)^2\) in exponential notation. Answer: The expression is already in exponential notation: \(a^3 - (b+7)^2\).

Step 1: Rewrite a cubed

To rewrite \(a\) cubed in exponential notation, we write \(a^3\).

Step 2: Rewrite the square of b + 7

To rewrite \((b+7)\) squared in exponential notation, we write \((b+7)^2\).

Step 3: Combine and simplify

Now that we have the exponential notation for each part of the expression, we combine them together using the subtracting operation: \(a^3 - (b+7)^2\). This is the final expression in exponential notation.

Key Concepts

Exponents

Exponents are a mathematical way to express repeated multiplication of a number by itself. For example, when we talk about "\(a\) cubed," we mean that \(a\) is multiplied by itself three times, which can be written in exponential notation as \(a^3\). This is the basis for understanding and using exponents in various algebraic contexts.
Exponents have two main parts:

• The base, which is the number being multiplied (\(a\) in our case).
• The exponent, which tells us how many times the base is used as a factor (3 for \(a^3\)).

Understanding exponents helps simplify expressions, as they allow complex multiplication processes to be expressed in a compact form. Learning to recognize, write, and work with exponential notation is essential in algebra and many other areas of mathematics.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(a\) or \(b\)), and operations (like addition or subtraction). They allow us to generalize mathematical problems and create "formulas" that can solve different instances of a problem. For instance, in the problem \((b+7)^2\), we have an expression that involves adding 7 to \(b\), then squaring the result.
Algebraic expressions can involve:

• Constants, which are numbers on their own.
• Variables, which are symbols representing numbers (e.g., \(a\), \(b\)).
• Operators, like addition, subtraction, multiplication, and division.

These expressions can become quite complex, especially when they include exponents. However, the use of exponential notation simplifies handling them by encapsulating repeated operations in an easy-to-read format.

Simplification

Simplification in mathematics means reducing an expression or equation to its simplest form without changing its value. It involves eliminating unnecessary parts and combining like terms, making the expression easier to understand and solve.
When we had \(a^3 - (b+7)^2\), simplifying involved ensuring that each part of the expression was correctly expressed in exponential notation. This did not involve further simplification by solving the expression, as it was already in its simplest exponential form.
The process of simplification often includes:

• Combining like terms (terms that have the same variables and exponents).
• Rewriting complex parts in simpler or more recognizable notations, such as \(a^3\) instead of saying "\(a\) cubed."
• Applying algebraic rules to reduce the complexity or length of expressions, keeping their value intact.

Mastering simplification helps in dealing with complex algebraic expressions efficiently, paving the way for solving equations easily and accurately.