Problem 79 - Elementary Algebra

Question

For the following problems, write the expressions using exponential notation. \((a+2 b)\) squared minus \((a+3 b)\) to the fourth.

Step-by-Step Solution

Answer

Question: Rewrite the given expression using exponential notation and simplify: \((a+2b)^2 - (a+3b)^4\) Answer: \((a+2b)(a+2b) - (a+3b)(a+3b)(a+3b)(a+3b)\)

Step 1: Identify the terms in the expression

The expression provided is: \((a+2b)^2 - (a+3b)^4\) Here, the two terms are \((a+2b)^2\) and \((a+3b)^4\).

Step 2: Apply the exponent rules to the individual terms

Now we will apply the exponent rules to both terms. For the first term, \((a+2b)^2\), squaring the term means multiplying it by itself: \((a+2b)^2 = (a+2b)(a+2b)\) For the second term, \((a+3b)^4\), raising the expression to the fourth power means multiplying it by itself four times: \((a+3b)^4 = (a+3b)(a+3b)(a+3b)(a+3b)\)

Step 3: Write the final expression

After applying the exponent rules, the final expression is: \((a+2b)(a+2b) - (a+3b)(a+3b)(a+3b)(a+3b)\) Now the given expression is written using exponential notation and represents the difference between \((a+2b)\) squared and \((a+3b)\) to the fourth power.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operators (such as addition, subtraction, multiplication, and division) that represent a specific value. In our problem, the expression is \((a+2b)^2 - (a+3b)^4\).This is a symbolic representation of what we will eventually calculate.

Algebraic expressions can be simple, such as \(x + 2\),or more complex, like the example we are exploring. The variables \(a\) and \(b\)represent unknown quantities, and the numbers attached to them are coefficients.

When dealing with algebraic expressions, here are a few key points to remember:

• Terms: Parts of the expression separated by addition or subtraction signs. For example, in the expression, \((a+2b)^2\) and \((a+3b)^4\) are two separate terms.
• Coefficients: These are the numerical parts that are multiplied by the variables. For \(2b\), 2 is the coefficient.
• Constants: Numbers without variables, but in our exercise, they are not present directly.

Understanding these components will allow you to simplify and manipulate expressions accurately.

Exponent Rules

Exponents are a shorthand notation to show how many times a number (the base) is multiplied by itself. When we deal with expressions in the form \((x^n)\),we use exponent rules to simplify and solve them. In our exercise, there are two such expressions: \((a+2b)^2\) and \((a+3b)^4\).

A basic rule of exponents is the power rule, which states that raising a power to another power involves multiplying the exponents. However, in this case, we simply multiply the base by itself as many times as stated by the exponent:

• In \((a+2b)^2\), it means \((a+2b)(a+2b)\),multiplying the base \((a+2b)\) by itself twice.
• In \((a+3b)^4\), it means \((a+3b)(a+3b)(a+3b)(a+3b)\),multiplying \((a+3b)\) by itself four times.

When you understand these exponent rules, you'll know how to tackle different algebraic expressions more efficiently.

Polynomials

A polynomial is a type of algebraic expression that includes terms composed of variables raised to whole number exponents and their coefficients. The expressions such as \((a+2b)^2\) and \((a+3b)^4\) are considered polynomials once expanded. Polynomials are named based on their degrees:

• Monomial: A polynomial with one term (like \(5x\)).
• Binomial: A polynomial with two terms (like \(x + 1\)).
• Trinomial: A polynomial with three terms (like \(x^2 + x + 1\)).

In this exercise, our terms \((a+2b)^2\) and \((a+3b)^4\) have not been expanded but they can become binomials or trinomials depending on how they are simplified.
Learning to expand and simplify polynomials is essential in algebra. This allows you to work with more complex equations and identify relationships between variables.